INVESTIGATING THE COUPLING BETWEEN TECTONICS,
CLIMATE AND SEDIMENTARY BASIN DEVELOPMENT
by
Todd M. Engelder
A Dissertation submitted to the faculty of the
DEPARTMENT OF GEOSCIENCES
In partial fulfillment of the requirements
For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
2012
THE UNIVERSITY OF ARIZONA
GRADUATE COLLEGE
As members of the Dissertation Committee, we certify that we have read the dissertation
prepared by Todd M. Engelder
entitled “Investigating the effects of climate, tectonics and sedimentary basin
development”
and recommend that it be accepted as fulfilling the dissertation requirement for the
Degree of Doctor of Philosophy
_______________________________________________________________________
Date: February 17, 2012
Jon Pelletier
_______________________________________________________________________
Date: February 17, 2012
Peter DeCelles
_______________________________________________________________________
Date: February 17, 2012
Paul Kapp
_______________________________________________________________________
Date: February 17, 2012
Clement Chase
_______________________________________________________________________
Date: February 17, 2012
Peter Reiners
Final approval and acceptance of this dissertation is contingent upon the candidate’s
submission of the final copies of the dissertation to the Graduate College.
I hereby certify that I have read this dissertation prepared under my direction and
recommend that it be accepted as fulfilling the dissertation requirement.
________________________________________________ Date: February 17, 2012
Dissertation Director: Jon Pelletier
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an
advanced degree at the University of Arizona and is deposited in the University Library
to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission, provided
that accurate acknowledgment of source is made. Requests for permission for extended
quotation from or reproduction of this manuscript in whole or in part may be granted by
the author.
SIGNED: Todd M. Engelder
4
ACKNOWLEDGEMENTS
I would like to start out by thanking my committee and especially my advisor Jon
Pelletier for helpful suggestions and guidance during my research. Finishing my PhD
would not have been possible without financial support from Exxon Mobil through the
COSA project. I thank my colleagues Dave Pearson, Jon Volkmer and Ross Waldrip for
fielding countless questions about structural geology when I couldn’t locate DeCelles or
Kapp. Thank you Caitlin Orem, Onne Crouvi and previous officemates for the many long
discussions we had about Geomorphology and life in general. Finally, thank you Josh
Spinler, Alexander Rohrmann, James Girardi, Andrew Kowler, Mark Warren, Meg
Blome, Mark Trees, Mauricio Ibanez-Mejia, Maria Banks and friends outside of Tucson
for helping me to “constructively” spend the few hours that I was not focusing on
geology.
5
DEDICATION
I dedicate this dissertation to my parents, Terry and Jan, and my sisters, Zoe and Stacey.
Thank you for all your visits, support and words of encouragement throughout graduate
school.
6
TABLE OF CONTENTS
LIST OF ILLUSTRATIONS……………………..……………………………. 9
ABSTRACT……………...………………………………………………………..... 10
INTRODUCTION………………………………………………………………… 12
PRESENT STUDY………………………………………...……………………... 22
REFERENCES………………..…………………………………………………... 31
APPENDIX A: STOCHASTIC VARIATIONS IN WATER FLOW
DEPTH AND THEIR ROLE IN LONG-RUNOUT GRAVEL
PROGRADATION IN SEDIMENTARY BASINS
…………………………………………………………………………………………. 40
Abstract……………………………………………………………………………... 40
1. Introduction…………………………………………………………………...…. 42
2. Numerical Model Description…………………………………………………... 46
3. Numerical Modeling Results……………………………………………….….... 58
4. Discussion………………………………………………………………….…….. 67
5. Conclusions………………………………………………………………………. 74
Acknowledgements……………………………………………………………….... 76
References…………………………………………………………………. ………. 76
Tables, Figures and Figure Captions……………………………………………... 86
APPENDIX B: SIMULATING FORELAND BASIN RESPONSE TO
MOUNTAIN BELT KINEMATICS AND CLIMATE CHANGE FOR
THE CENTRAL ANDES: A NUMERICAL ANALYSIS OF THE
CHACO FORELAND IN SOUTHERN BOLIVIA
…………………………………………………………………………………………. 91
Abstract………………………………………………………………………...…… 91
7
1. Introduction……………………………………………………..……….………. 93
2. Geologic Background………………………..…………...……………………... 100
3. Numerical Model Description…………………………………………………... 103
3.1. Deformation Model……………………………………………………...….. 104
3.2. Erosion Model…………………………………………………………...….. 105
3.3. Sediment Transport Model…………………………………………….…... 106
3.4. Basin Flexure ……………………………………………………………..... 108
3.5. Numerical Modeling Methods……………………………………………... 109
4. Numerical Modeling Results…………………………………………………..... 113
4.1. End-member surface-uplift model experiment summary……….……..… 113
4.2. Constraints on foreland basin depositional rates………………………..... 117
4.3. The role of eclogite-root-foundering on surface uplift and foreland
development……………………………………………………………………… 120
4.4. The role of climate change on the foreland development………………… 123
5. Discussion……………………………………………………………………..…. 125
6. Conclusions………………………………………………………………………. 133
Acknowledgements………………………………………………………………… 135
References………………………………………………………………………….. 135
Tables and Figures………………………………………………………………..... 148
APPENDIX C: QUANTIFYING THE EFFECT OF HYDROLOGIC
VARIABILITY ON BEDLOAD SEDIMENT TRANSPORT IN
ALLUVIAL CHANNELS
…………………………………………………………………………………………. 156
Abstract…………………………………………………………………………...… 156
1. Introduction…………………………………………………………………...…. 158
2. Analytic Solutions……………………………………………………………..… 164
2.1. Derivation of an analytic equation for long-term bedload sediment flux
……………………………………………………………………………………. 164
2.2. Derivation of the diffusivity coefficient…………………………….…….... 179
2.3. Numerical and analytic solutions for the effective discharge……..……... 181
8
3. Preliminary data analysis required for the validation of model predictions
………………………………………………………………………...…………….. 182
3.1. Climate effect on long-term bedload sediment flux………………………. 182
4. Methods of Sensitivity Studies………………………………………………….. 187
4.1. Diffusivity sensitivity studies……………………………………………......187
4.2. Long-term bedload sediment transport sensitivity studies………………. 189
4.3. Effective discharge return period sensitivity studies……………………... 192
5. Results of the Sensitivity Studies……………………………………………….. 193
5.1. Diffusivity sensitivity studies……………………………………………..... 193
5.2. Climate effects on long-term sediment transport………………………… 196
5.3. Climate effects on effective discharge return period…………………….. 202
6. Discussion………………………………………………………………………... 207
6.1. Diffusivity…………………………………………………………………… 207
6.2. Climate effects on sediment transport and effective discharge return period
……………………………………………………………………………………. 208
7. Conclusions………………………………………………………………………. 214
Notation…………………………………………………………………………….. 216
References………………………………………………………………………….. 218
Tables……………………………………………………………………………...... 225
Figures………………………………………………………………………………. 227
9
LIST OF ILLUSTRATIONS
Figure 1: Conceptual diagrams of a gravel wedge prograding………………………… 30
Figure 2: A plot of the relationship between long-term sediment transport, discharge
frequency and sediment transport rates…………………………………………………30
10
ABSTRACT
Sedimentary deposits have been broadly used to constrain past climate change
and tectonic histories within mountain belts. This dissertation summarizes three studies
that evaluate the effects of climate change and tectonics on sedimentary basin
development. (1) The paleoslope estimation method, a method for calculating the
threshold slope of a fluvial deposit, does not account for the stochastic variations in water
depth in alluvial channels caused by climatic and autogenic processes. Therefore, we test
the robustness of applying the paleoslope estimation method in a tectonic context. Based
on our numerical modeling results, we conclude that if given sufficient time gravel can
prograde long distances at regional slopes less than the minimum transport slope
calculated with the paleoslope estimation method if water depth varies stochastically in
time, and thus, caution should be exercised when evaluating regional slopes measured
from the rock record in a tectonic context. (2) The role of crustal thickening, lithospheric
delamination, and climate change in driving surface uplift in the central Andes in
southern Bolivia and changes in the creation of accommodation space and depositional
facies in the adjacent foreland basin has been a topic of debate over the last decade. Our
numerical modeling results show that gradual rise of the Eastern Cordillera above 2-3 km
prior to 22 Ma leads to sufficient sediment accommodation for the Oligocene-Miocene
foreland basin stratigraphy, and thus, the Eastern Cordillera gained the majority of its
modern elevation prior to 10 Ma. Also, we conclude that major changes in grain size and
depositional rates are primarily controlled by mountain-belt migration (i.e., climate
change and lithospheric delamination are secondary mechanisms). (3) Existing equations
11
for predicting the long-term bedload sediment flux in alluvial channels include mean
discharge as a controlling variable but do not explicitly include variations in discharge
through time. We develop an analytic equation for the long-term bedload sediment flux
that incorporates both the mean and coefficient of variation of discharge. Our results
show that although increasing aridity leads to an increase in large discharges with respect
to small discharges, long-term bedload sediment transport rates decrease for both gravel
and sand-bed rivers with increasing aridity.
12
INTRODUCTION
The stratigraphy of sedimentary basins is a partial record of climate change and
tectonics because the fluvial systems that convey sediment from the bedrock drainages
into the depositional basins are sensitive to changes in discharge, sediment texture,
sediment supply from adjacent bedrock drainages and sediment accommodation rates.
Thus, sedimentary deposits have been broadly used to constrain past climate change and
tectonic histories within mountain belts (e.g. Heller et al., 1988; DeCelles et al., 1998;
Gaupp et al., 1999; Marzo and Steel, 2000, Uba et al., 2007). However, linking
stratigraphic patterns within sedimentary basin stratigraphy to either climate change or
tectonics alone is challenging because the stratigraphy is a partial signal of both of these
processes as well as autogenic processes (e.g. entrenchment and filling cycles and
channel avulsion). For example, the onset of the South American monsoon, mountainbelt migration and rapid surface uplift within the mountain belt are three mechanisms that
have been proposed for causing the grain size and depositional rate increases within the
Neogene stratigraphy of the Chaco Foreland basin of the central Andes in southern
Bolivia (DeCelles and Horton, 2003; Uba et al., 2006; Uba et al., 2007). This dissertation
contains three studies that aim to further improve upon our understanding of how climate
change, tectonics and autogenic processes affect the stratigraphy of and the sediment
transport rates within sedimentary basins.
Several studies have used the paleoslope estimation method developed by Paola
and Mohrig (1996) combined with grain size and channel depth data from gravel units
that were deposited in alluvial channels to infer the presence or absence of tectonically-
13
driven tilting since the deposition of the gravel units (e.g. Kirby et al., 2000; McMillan et
al., 2002; Cassel and Graham, 2011). It has been long recognized that sediment grains
require a critical shear stress to initiate entrainment (Shields, 1936; Meyer-Peter and
Mueller, 1948). Using the concept of a critical shear stress, Paola and Mohrig (1996)
showed that the minimum transport slope for gravel-bed channels with cohesionless
channel banks is proportional to bankfull flow depth and the median grain size of the
channel bed sediment. Therefore, the stratigraphy of depositional units can be analyzed in
outcrops to obtain these parameters and calculate the minimum paleoslope at which the
coarse sediments within the depositional units were transported. However, the paleoslope
estimation method does not account for climatically and autogenically driven changes in
water depth that occur within alluvial channels, and thus, the paper in Appendix A
evaluates the robustness of using the paleoslope estimation method to infer tectonics
within sedimentary basins through a numerical modeling analysis.
Specifically, we focus on the study in which Heller et al. (2003) calculated
paleoslopes for three long-runout gravel units in the Western U.S. (i.e., Shinarump
Conglomerate of the Upper Triassic Chinle Formation, Lower Cretaceous conglomerate
units and the Tertiary Ogallala Group). Long-runout gravels as defined by Heller et al.
(2003) are gravels that have been transported hundreds of kilometers from their source
regions. Heller et al. (2003) found that the calculated minimum paleoslopes for each of
the three gravel units are greater than the slopes measured in outcrop. The three longrunout gravels were deposited on top of sedimentary units that contain depositional faces
indicative of low-energy environments, and therefore, Heller et al. (2003) inferred that
14
topography was originally flat at the beginning of gravel transport and deposition. In the
absence of tectonics, the clastic wedge created by gravel deposition would have to
maintain the minimum transport slope to convey gravels far out into the basin (Figure
1A). However, the slope that Heller et al. (2003) measured from the internal relief of the
gravel deposits was less than the calculated minimum transport slope. Based on this
result, Heller et al. (2003) concluded that tectonic tilting must have occurred to allow
gravels to be transported hundreds of kilometers from their source region (Figure 1B).
A key assumption of the paleoslope estimation method is that the fining upward
sequence measured in outcrops represents the flow depth during gravel transport.
However, flow depth varies in space and time in ways that may affect the accuracy of the
paleoslope estimation method. For example, flow depth varies in response to one to
several year changes in flood discharge and at longer time scales in response to climatic
and autogenic processes (Schumm and Hadley, 1957; Leopold, 1964; Patton and
Schumm, 1975). The non-steady-state nature of channel geometries is also significant
because the stratigraphic record contains only depositional channels within the infill (i.e.,
only the final stage of active-incision is preserved as a scour surface). As such, the
paleoslope estimation method uses an inherently biased proxy for flow depth during
transport of the coarsest load if channel dimensions are measured from an individual
stacked channel within an infilling succession. Given these considerations, it is
reasonable to expect that threshold slopes for entrainment in natural rivers, which are
function of flow depth, vary stochastically in time. If the threshold slope of entrainment
varies stochastically through time, then the transport of coarse sediments such as gravel is
15
no longer controlled by just an average threshold condition (i.e., the threshold inferred
from measuring the flow depth from outcrop). Instead, gravel may be able to prograde at
channel slopes lower than the minimum value predicted by paleoslope estimation theory,
albeit for geologically brief periods of time when threshold slope is lower than average.
The paper contained in Appendix B uses a coupled mountain-belt and foreland
basin numerical model to constrain the surface uplift history of the eastern margin of the
central Andes in southern Bolivia as well as constrain the effects of climate change and
continental lithospheric delamination on the foreland basin stratigraphy. Numerical
models are useful tools for simulating the response of alluvial channel profiles, and thus
sedimentary basins, over geologic time scales to climatic and tectonic forcings. The first
attempts to quantitatively predict sedimentary basin subsidence came from McKenzie
(1978) and Beaumont (1978). The Beaumont (1978) study showed that stratigraphic
patterns within foreland basins might contain a signature of lithospheric rheology (Paola,
2000). Later, Flemings and Jordan (1989) was the first study to couple subsidence and
surface processes in a 2-dimensional foreland basin model. In their model, Flemings and
Jordan (1989) simulated erosion and sediment transport using a diffusion model, which
assumes that sediment flux is linearly proportional to the surface slope. Although the rate
of sediment transported by an individual flood can be highly non-linear with respect to
the channel slope, the long-term sediment transport rates averaged over many flood
events has been both experimentally and theoretically shown to approximate diffusion
(Begin et al., 1981; Paola et al., 1992). The results of Flemings and Jordan (1989) showed
that the growth of a mountain belt eventually causes foreland basins to switch from
16
underfilled to overfilled as drainage development and increasing mountain front slopes
lead to sediment input rates surpassing sediment accommodation rates. A second result of
the study was that depositional energy, and thus, grain size is primarily influenced by a
depositional zone’s distance from the approaching mountain front. Following these early
studies, coupled numerical models have improved by adding complexity to surface
processes, dynamically predicting lithospheric deformation or simulating evolution of
foreland basins in three dimensions (e.g. Coulthard et al., 2002; Clevis et al., 2003;
Simpson, 2004).
An unresolved issue for the central Andes in southern Bolivia that is addressed in
a numerical modeling study described in Appendix B is: when did the topography of
central Andes rise to its modern elevation? The classic model for the central Andes poses
that the majority of surface uplift occurred in response to Neogene thermal weakening of
the lithosphere and deformation within the Subandean zone (a regional map that shows
the location of tectonomporphic zones can be found in Figure 1 of Appendix B), which is
the modern fold-and-thrust belt, and that earlier Tertiary deformation contributed to a
small fraction of the modern topography (Isacks, 1988). Results from paleoelevation and
geomorphic studies indeed support Late Neogene rapid surface uplift of the region
(Gubbels et al., 1993; Barke and Lamb, 2006; Ghosh et al., 2006; Hoke et al., 2007;
Garzione et al., 2008). A second conceptual surface uplift model poses that the central
Andes have been gradually rising since deformation propagated east into the Eastern
Cordillera and that the Eastern Cordillera had risen to near modern elevation prior to the
Late Neogene. Evidence for pre-Neogene deformation and surface uplift come from
17
stratigraphic and thermochronologic studies (Horton et al., 2002; McQuarrie et al., 2005;
Ege et. al., 2007; Barnes et al., 2008).
A second goal of the paper contained in Appendix B is to determine if the
stratigraphic pattern of the Neogene stratigraphy of the Subandean zone is primarily
controlled by the eastward propagation of the mountain belt or if there is a signal of
climate change and/or continental lithospheric delamination. The Petaca Formation
located at the base of the Neogene section contains fluvial deposits that are characterized
by well developed paleosols and westward paleocurrent directions (Uba et al., 2006).
Above the Petaca Formation, the mean grain size decreases and transport directions are
predominantly north-south in the Yecua Formation. The transition upward into the
Tariquia Formation occurs as a factor of five increase in depositional rates, grain size
increase and a shift in paleocurrent directions toward the east. DeCelles and Horton
(2003) and Uba et al. (2006) interpreted the Tertiary stratigraphy as evidence for a
migrating mountain belt. However, Uba et al. (2007) interpreted the change in
depositional rates and depositional facies between the Yecua and Tariquia Formation as a
signal of climate change. Kleinert and Strecker (2001) documented a change from
previously dry to wetter conditions in the Santa Maria basin of Northern Argentina
between 9-7 Ma. Uba et al. (2007) posed that the onset of the South American monsoon
would lead to higher sediment input from the mountain belt, and thus, higher depositional
rates within the foreland basin.
Continental lithospheric delamination is another mechanism that may have
influenced the stratigraphy of the Chaco foreland basin at this time. Continental
18
lithospheric delamination involves the removal of negatively buoyant lower crust (i.e.,
eclogite root) and mantle lithosphere. DeCelles et al. (2009) pose that in cordilleran
orogenic systems like the Andes large volumes of lower crust are thrust beneath the
developing orogen. Through melting of the lower crust beneath the magmatic arc, the
lower crust is partitioned into a felsic melt and a mafic residual. The mafic residual can
form a dense eclogitic root that is negatively buoyant with respect to the surrounding
mantle if it is formed at pressures that exceed ~15 kbar (Wolf and Wyllie, 1993; Rapp
and Watson, 1995; Rushmer, 1995). The eclogite root continues to grow as lower crust is
thrust beneath the magmatic arc until it reaches a critical mass and founders into the
mantle. The growth and foundering of an eclogite root is part of a conceptual model
called the Cordilleran Cycle that relates feebacks between convergence rates, magmatic
processes and delamination (DeCelles et al., 2009). Growth and delamination of an
eclogite root has consequences for the foreland basin stratigraphy because the eclogite
root is a subsurface load that modifies the elevation of the overlying mountain belt, and
thus, the deflection of the lithosphere underlying the foreland basin. The
presence/absence of an eclogite root also affects shortening rates within the upper plate,
and thus, affects the rate of mountain belt propagation into the foreland basin. Thus,
unconformities, changes in grain size and/or changes in depositional rates within the
Neogene stratigraphy of the Subandean Zone of the central Andes may be a partial signal
of eclogite root growth and delamination.
Thus far we have discussed sedimentary basins that are partially influenced by
tectonics or feedbacks with the basin-bounding topography. However, the paper included
19
in Appendix C focuses only on climate change as the mechanism for varying long-term
sediment transport rates in sedimentary basins. Long-term in the context of this research
refers to time scales of decades to centuries, i.e. sufficiently long that the estimated
sediment flux for a given location includes the cumulative effects of many flood events
but not so long that the estimate averages over the effects of different climatic conditions.
Constraining the effect of climate change on alluvial channel bedload transport efficiency
is necessary for understanding the evolution of sedimentary basins, because upstream
sediment flux acts as a principal boundary condition for terrestrial and marine depozones
(Paola et al., 1992; Tucker and Slingerland, 1999; Molnar, 2004; Barnes and Haines,
2009). Changes in transport efficiency within a sedimentary basin can also modify
erosional rates in the upstream drainage basins because the slope of the sedimentary basin
sets the local base level of the drainage basin.
An unresolved question in fluvial geomorphology posed by Molnar (2001) is:
does increasing aridity lead to an increase in the long-term sediment transport of a
channel? Long-term sediment transport is the total sediment transported by distribution of
floods or discharges that occur in a channel. Two important parameters of the discharge
distribution are the mean and variation. Analysis of channel hydrographs reveals that
discharge variability, expressed as a coefficient of variation (i.e., the standard deviation
normalized by the mean), increases with increasing aridity. Turcotte and Greene (1993),
for example, analyzed peak annual discharge for 10 channels in various climates across
the United States. Their results showed that the ratio of the frequency of the largest
discharges to that of the mean discharge increased significantly with increasing aridity. In
20
the most comprehensive analysis to date, McMahon et al. (2007) analyzed the sample
statistics of 1221 channels globally. Their results confirm a negative correlation between
the coefficient of variation of annual discharge and mean annual runoff. Therefore, as
climates shift to greater aridity (i.e., decrease in the mean discharge), the variability in
discharge increases. Discharge variability is important for long-term transport rates
because sediment transport is a nonlinear function of discharge. As such, Molnar (2001)
concluded that channels in arid climates would transport more sediment than the same
channel in a humid climate. In effect, Molnar (2001) argued that the increase in the
flashiness of discharge more than compensated for the decrease in mean discharge with
increasing aridity. The reason for this is related to the fact that large floods are so much
more effective, per unit discharge, than small floods.
Another unresolved question in fluvial geomorphology that is addressed in
Appendix C is: what is the effect of increasing aridity on the return period of the effective
discharge. The effective discharge is defined as the discharge magnitude that transports
the most sediment over an interval of time (Wolman and Miller, 1960). The return period
of the effective discharge has implications for the stability of a channel. Channels that
have effective discharges with long return periods are thought to spend the majority of
their time in a state of recovery. Another implication for return period of the effective
discharge is to assess the time scales over which modern bedload fluxes are
representative of the bedload component of long-term basin-averaged erosion rates. In a
study of mixed bedrock and alluvial drainages in Idaho, Kirchner et al. (2005) found that
basin-average erosion rates representing time scales of 103 to 107 years were an order of
21
magnitude greater than that those that represent 1 to 10 years. The decrease in basinaveraged erosion rates was interpreted as an effect of the sampling period, and thus, the
modern erosion rates had not yet sampled a thousand year event that does more than 90
percent of the long-term geomorphic work.
22
PRESENT STUDY
The methods, results, and conclusions of this study are presented in the papers appended
to this dissertation. The following is a summary of the most important findings in this
document.
Previous studies have used the paleoslope estimation method of Paola and Mohrig
(1996) to calculate the minimum paleoslopes of fluvial systems that deposited longrunout gravels in the western U.S. (McMillan et al., 2002; Heller et al. 2003). Heller et al.
(2003) found that the upper slopes of the gravel wedges for each long-runnout gravel due
to deposition above previously flat lying sediments were less than the calculated
minimum slope required to entrain the gravel. Heller et al. (2003) interpreted the low
gradients of the gravel deposits as evidence of tectonically driven regional tilting. For
example, Heller et al. (2003) proposed that the long-wavelength tilting that led to the
transport of the Miocene Ogallala Group into the Great Plains of the Western U.S. was
caused by a combination of dynamic rebound due to the passage of the Farallon slab and
regional lithospheric warming associated with extension within the Rio Grande Rift.
Using a 2-D coupled mountain belt and foreland basin numerical model, we
explore the following question: given a sufficient period of time, can gravels prograde
several hundred kilometers from their source regions at slopes significantly lower than
the threshold slope predicted by the paleoslope estimation method given realistic
variations in the threshold slope of entrainment in space and/or time (Appendix A)? A
key assumption of the paleoslope estimation method is that the fining upward sequence
23
measured in outcrop is a proxy for the flow depth during gravel transport. However, flow
depth varies in space and time due to climatic and autogenic processes such that gravel
transport is a function of the variation in flow depth in addition to the mean flow depth
and median grain size. In order to test the hypothesis that gravels can be transported long
distances at regional slopes below the threshold slope predicted by the paleoslope
estimation method, we allow threshold slopes that are a function of flow depth and
median grain size to vary stochastically through time in our numerical model. Stochastic
variation in threshold slope is simulated with a lognormal distribution. The coefficient of
variation of the lognormal distribution was calculated using data from North American
gravel-bed rivers (Church and Rood, 1983).
Simulations were run for two types of sedimentary basins: (1) isolated
sedimentary basin and (2) a postorogenic foreland basin. The isolated sedimentary basin
model allows us to isolate the effects of stochastic variation in threshold slopes on gravel
transport independent of tectonics. The results of the isolated basin model show that
gravel can prograde at regional slopes below the minimum transport slope predicted by
the paleoslope estimation method if the coefficient of variation of threshold slope is
greater than zero and the sediment supply does not greatly exceed the transport capacity
of the main gravel-bed rivers within the depositional basin. A comparison of this result
with the dynamically-coupled postorogenic foreland basin model results allows us to
determine whether or not feedbacks between sediment supply to the basin and the base
level of erosion for the mountain belt (controlled by transport efficiency of sediments
across the basin) act in concert to control the rates and slopes at which a gravel wedge
24
can prograde within a sedimentary basin. The time required for the regional slope of the
gravel-bed rivers to decrease below the minimum transport slope predicted by the
paleoslope estimation method is depend on the transport efficiency of the rivers, the
sediment supply rates from the upstream drainage basins and the variability of the
threshold slope. The results of the postorogenic foreland basin model show that soon after
the cessation of active uplift in the mountain belt (< 2 Myr), the regional slopes of gravelbed rivers are generally high because sediment fluxes from the mountain belt are high.
Beyond 2 Myr, the regional slopes of the gravel-bed rivers begin to decrease as sediment
flux from the mountain belt decreases and the distal toe of gravel depositional passes into
the postorogenic backbulge basin.
In our second study (Appendix B), a single 2-D coupled mountain belt and
foreland basin numerical model was used to explore three issues for the central Andes:
(1) constrain the surface uplift history of the eastern margin of the central Andes through
comparing the results of end-member surface uplift experiments against the observed
foreland basin stratigraphy, (2) constrain the effect of the continental lithospheric
delamination on the foreland basin stratigraphy and (3) constrain the effect of climate
change on the foreland basin stratigraphy. For the end-member uplift model experiments,
climate and the rigidity of the South American plate are held constant. The timing of
initiation and duration of crustal deformation within each of the tectonomorphic zones
(e.g. Eastern cordillera, Interandean zone, Subandean zone) was prescribed based on
published exhumation ages and stratigraphic relationships (Muller et al., 2005; Ege et al.,
2007; Barnes et al., 2008). Rock uplift rates and bedrock erodibilities were calibrated in
25
the end-member surface uplift model experiment by fitting the observed modern
topography and exhumation magnitudes. The results of this experiment show that the
gradual uplift model fits the Eocene-early Miocene foreland basin stratigraphy best, and
therefore, the Eastern Cordillera should have gained significant elevation (i.e., >2 km) by
22 Ma. Although the Eastern Cordillera was near modern elevation by 10 Ma, rapid uplift
within the Interandean and western Subandean zones is still necessary to create sufficient
sediment accommodation space for the Late Miocene foreland basin stratigraphy.
The continental lithospheric delamination experiments involved the growth and
delamination of an eclogite root. The eclogite root was prescribed to be 100 km wide and
located within the eastern Altiplano and Eastern Cordillera backthrust region based on
geophysical data from the central Andes of Southern Bolivia (Beck and Zandt, 2002). We
assume that the eclogite delamination is caused by a Raleigh-Taylor instability (defined
as the diapiric drip of a dense layer overlying a less dense layer). We also assume that the
majority of the delamination event is spent growing the instability (i.e., drip) by a factor
of e, and thus, the maximum thickness of the eclogite root prior to delamination can be
calibrated such that the majority of the delamination period was spend growing the
instability. As such, we prescribe the thickness of the eclogite layer prior to delamination
to be approximately 12.5 km. In this experiment the climate, elastic rigidity, timing and
duration of deformation are the same as in the end-member surface uplift experiment.
Between the initial
The results of the continental lithospheric delamination experiment show that the
delamination of an eclogite root of the prescribed dimensions (i.e., 100 by 12.5 km)
26
causes approximately 1 km of isostatic rebound at the boundary between the Altiplano
and Eastern Cordillera directly above the center of the root when the rigidity of the South
American plate is approximately 6.8x1023 Nm. The isostatic rebound is not uniform,
however, and rapidly decreases away from the center of the eclogite root. During the
development of the foreland basin between 30 and 7 Ma the foredeep is always more than
300 km away from the center of the eclogite root. The deflection of the foredeep due to
the growth and delamination of the eclogite root is less than or equal to 100 meters, and
thus, the depositional rates of the foredeep stratigraphy is expected to be more sensitive
to surface uplift rates in the front of the mountain belt than growth and removal of an
ecologite root located on the eastern edge of the Altiplano. However, the growth and
removal of an eclogite root may have a more significant effect on the foreland basin
stratigraphy if the presence or absence of an eclogite root significantly modifies the
convergence rates between the mountain belt and the South American plate (DeCelles et
al., 2009).
The final experiment involved constraining the effect of the onset of the South
American monsoon on the Late Miocene foreland basin stratigraphy. In this experiment
the both erosion rates and sediment transport rates are reduced by a factor of two with
respect to the values used within the end-member surface uplift models between 43 and 9
Ma. At 9 Ma, the erosion and sediment transport rates are increased by a factor of two to
simulate the onset of the South American monsoon.
The results of the climate change experiment show that sediment fluxes from the
central Andes into the foredeep basin are sufficient to overfill the basin from the Miocene
27
to present, and thus, sediment accommodation is the dominant mechanism for controlling
depositional rates within the foreland basin.The results for the climate change experiment
also show that increasing erosion and precipitation rates by a factor of two has a less
dominant effect on the distribution of grain size within the foreland basin than migration
of the mountain belt toward the foreland basin, and thus, is in agreement with the results
of Flemings and Jordan (1989).
In the third study (Appendix C), we derive an expression for the long-term
bedload sediment transport rates of alluvial channels and use the expression to explore
three applications: (1) constrain the effect of climate change on long-term bedload
sediment transport rates, (2) derive a diffusivity that includes the effects of discharge
variability and (3) constrain the effect of climate change on the effective discharge.
Wolman and Miller (1960) first proposed that the geomorphic work (i.e. long-term
sediment transport) performed by a channel could be quantified by integrating the
product of a sediment transport function and a frequency-size distribution of discharge
over the range of all possible discharges (Figure 2). Thus, we integrated the bedload
sediment transport formula of Wiberg and Smith (1989) and Engelund and Hansen
(1967) with a lognormal distribution to solve for the long-term bedload sediment
transport rates for both gravel and sand-bed channels. In the first application we compare
gravel and sand-bed channels with the same grain size distributions and along profile
slopes with different mean annual runoffs to constrain the effect of climate on long-term
bedload sediment flux. Previous studies have demonstrated that discharge variation (i.e.,
the coefficient of variation of annual discharge) negatively correlates with mean annual
28
runoff and mean annual precipitation (Turcotte and Greene, 1993; Molnar et al., 2006;
McMahon et al., 2007). We analyze 530 hydrographs from North American channels to
develop an equation that relates the coefficient of variation of daily discharge to mean
annual runoff. The results of the first application show that long-term bedload sediment
transport rates decrease with increasing aridity for both gravel and sand-bed channels.
Although the frequency of large discharges increases with aridity, the threshold for
transporting both gravel and sand-sized grains is sufficiently low that shifting the mode
of the discharge distribution toward smaller discharges decreases the long-term bedload
sediment transport rate.
Another goal of the study described in Appendix C is to constrain the effect of
climate change on the effective discharge for bedload transport in both gravel and sandbed channels. The effective discharge is defined as the discharge magnitude that
transports the most sediment per time (Wolman and Miller, 1960). Wolman and Miller
(1960) showed that the effectiveness function for a given discharge distribution is the
product of the sediment transport function and the discharge distribution (Figure 2). The
discharge located beneath the peak of the effectiveness curve is the effective discharge.
The effective discharge can be calculated by setting the derivative of the effectiveness
curve equal to zero and then solve for discharge. Again we apply a lognormal distribution
and the bedload sediment transport formula of Wiberg and Smith (1989) and Engelund
and Hansen (1967) in this analysis. The results of this analysis show that the effective
discharge for bedload sediment transport within a sand-bed channel has a return period
that is less than a year when the mean annual runoff ranges between 0.01 and 1.0 m.
29
However, the return period for the effective discharge in gravel-bed channels can be
significantly longer (up to hundreds of years) when the mean annual runoff is less than
0.02 m. When the mean annual runoff is greater than 0.1 m, the return period of the
effective discharge for gravel-bed channels is less than or equal to one year. This result
has implications for how well the bedload component of modern basin-averaged erosion
rates represents the bedload component of erosion rates over thousands of years. Kirchner
et al. (2001) found that modern basin-averaged erosion rates for gravel-bed channels in
Idaho with mean annual runoffs of approximately 0.5m were an order of magnitude less
than erosion rates that represent time scales of 103-107 of years. Conversely, our results
show that the effective discharge for gravel-bed channels with mean annual runoffs
greater than 0.2m occur on average once a year, and thus, the modern long-term bedload
sediment transport rates measured in gravel-bed channels should reflect bedload transport
rates representative of greater time scales.
30
Figure 1: Conceptual diagrams of a gravel wedge prograding above (A) older flat-lying
sedimentary deposits and (B) tectonically-tilted older deposits in response to an upstream
sediment source. The regional slope of the gravel wedge due to deposition as measured
by Heller et al. (2003) is represented by the angle between the upper surface of the gravel
wedge and the underlying depositional unit.
Figure 2: A plot of the relationship between long-term sediment transport, discharge
frequency and sediment transport rates after Wolman and Miller (1960). The gray area,
which represents the long-term sediment transport rate, is bound on top by the
effectiveness curve of Wolman and Miller (1960).
31
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40
APPENDIX A: STOCHASTIC VARIATIONS IN WATER FLOW
DEPTH AND THEIR ROLE IN LONG-RUNOUT GRAVEL
PROGRADATION IN SEDIMENTARY BASINS
Manuscript in review with Lithosphere
Todd M. Engelder* and Jon D. Pelletier
Department of Geosciences, University of Arizona, 1040 E. Fourth St., Tucson AZ,
85721, USA,
*Corresponding Author, email: engelder@email.arizona.edu, fax: (520) 621-2672
Abstract
Several recent studies have used a threshold shear stress criterion, together with fieldbased measurements of median grain size and channel depth in alluvial gravel deposits, to
calculate the threshold paleoslopes of alluvial sedimentary basins in the western United
States. Threshold paleoslopes are the minimum slopes that would have been necessary to
transport sediment in those basins. In some applications of this method, inferred threshold
paleoslopes are sufficiently steeper than modern slopes that they imply large-magnitude
tectonic tilting must have occurred in order to transport sediments to their present
locations. In this paper, we evaluate the accuracy and robustness of the paleoslope
estimation method and we evaluate the conditions under which long-runout gravels (i.e.
gravels several hundred kilometers or more from their source regions) can occur using
numerical models for two types of sedimentary basins: (1) an isolated sedimentary basin
with a prescribed source of sediment from upstream, and (2) a basin dynamically coupled
to a postorogenic mountain belt. Specifically, we address the question: can long-runout
gravels prograde at regional slopes significantly lower than those predicted by the
paleoslope estimation method? In the models, threshold slopes for entrainment are varied
41
stochastically in time with an amplitude comparable to those of natural rivers in order to
represent fluctuations in flow depth due to hydroclimatic variability and local deviations
from an equilibrium channel geometry. The models show that when local threshold slope
values vary stochastically, alluvial basin sediments can persistently prograde at slopes far
below the threshold slopes predicted by paleoslope estimation theory. As such, the
models suggest that long-runout gravels do not require steep regional slopes in order for
transport to occur. We conclude that the minimum progradational slopes of fluvial
sedimentary basins adjacent to postorogenic mountain ranges are functions of the density
and texture of the bed sediment and both the mean and coefficient of variation of flow
depths associated with hydroclimatic variability and changes in local channel geometry
arising from autogenetic and allogenetic variations.
Keywords- sediment transport, foreland basin, isostatic rebound, stochastic processes
42
1. Introduction
The concept that gravel transport in rivers requires a threshold bed shear stress
has been recognized since the early 1900s (Shields, 1936; Meyer-Peter and Mueller,
1948). If flow conditions produce a shear stress that is greater than the threshold shear
stress required for entrainment, sediment transport will take place. Paola and Mohrig
(1996) used the threshold shear stress concept to relate the minimum channel slope
required for sediment transport to the median grain size and flow depth in gravel-bed
channels:
Sc=ψD50H-1
(1)
where Sc is the threshold channel slope, D50 is the median grain size of the mobile bed
sediments, H is the flow depth, and ψ (calculated to be 0.094 when the input grain size is
the D50) is a global constant for all gravel-bed channels with non-cohesive banks.
Equation (1) assumes that flow is quasi-steady, bedforms are not present, and that
channel banks are non-cohesive. Gravel-bed channels with non-cohesive banks modify
their cross-sectional geometries (e.g. channel width-to-depth ratio) to produce a bank
shear stress that is nearly equal to the threshold of entrainment (Parker, 1978; Andrews,
1984), hence channel width can be eliminated from the analysis that leads to equation (1).
When this relationship was tested against a dataset of modern gravel-bed rivers compiled
by Church and Rood (1983), a significant amount of scatter was observed in the ratio
between threshold slopes calculated with equation (1) and observed channel slopes (i.e.,
Sc/Sobs). To within one standard deviation of the geometric mean, predicted threshold
43
slopes for channels with non-cohesive banks were up to a factor of 1.72 both greater and
lesser than the observed slopes (Paola and Mohrig, 1996).
Equation (1) has recently been used to constrain the paleotopography of graveldominated alluvial deposits in the North American Cordillera (McMillan et al., 2002;
Heller et al., 2003). These deposits are considered to be long-runout gravels because they
extend several hundred kilometers from the bedrock drainage basins from which they
were sourced. Heller et al. (2003) calculated paleoslopes based on measured median grain
sizes and proxies for mean channel depth within outcrops of the Shinarump
Conglomerate of the Upper Triassic Chinle Formation, Lower Cretaceous conglomerate
units (e.g. Buckhorn Member of the Cedar Mountain Formation in central Utah and the
Cloverly Formation in Wyoming) and the Tertiary Ogallala Group. The channel slopes
calculated from equation (1) were significantly greater than the slope of the upper surface
of the clastic wedge that formed through deposition of gravel above previous topography
that Heller et al. (2003) interpreted to be nearly horizontal in slope based on depositional
facies indicative of low transport energy. As such, Heller et al. (2003) inferred that preor syn-depositional tectonic tilting must have taken place in order to produce the slopes
necessary to entrain sediment of the textures observed in these deposits.
A key assumption of the paleoslope estimation method, i.e. using equation (1) to
calculate threshold paleoslopes, is that the flow depth inferred from the thickness of the
fining-upward sequence in outcrop is equivalent to the minimum effective flow depth
during transport. This assumption allows mean data measured from outcrops to be used in
place of flow depth H in equation (1). However, flow depth varies in space and time in
44
ways that may affect the accuracy of the paleoslope estimation method. First, channels
are subject to discharges of varying magnitude. Hydrologic time-series data collected
within the last hundred years show that river discharges averaged on an annual basis can
follow normal, lognormal and power-law distributions with parameters that depend on
drainage basin area and climate (e.g. Leopold, 1964; Turcotte and Greene, 1993). As
discharge varies, the flow depth must also vary unless the channel width and slope
respond instantaneously to changes in shear stress, which they do not. If flow depth
varies and grain size does not respond instantly to maintain the threshold condition
defined by equation (1), then the threshold slope calculated by equation (1) must vary in
time.
At time scales longer than one to several years, threshold slopes also vary because
alluvial channels are non-steady-state systems. Instead, they often experience long-lived
(i.e., millennial time scales and greater) cut-and-fill cycles triggered both autogenically
(Schumm and Hadley, 1957; Patton and Schumm, 1975) and by climatic changes (Bull,
1997). The “cutting” cycle of alluvial channels is often accompanied by local channel
narrowing as unit stream power, incision, and channel narrowing operate in a strong
positive feedback. Conversely, depositional channels commonly widen as they aggrade.
Natural channels can oscillate between these two modes indefinitely, transporting the
coarsest load only during episodes of active channel incision when the channel is far from
equilibrium (Pelletier and DeLong, 2004). In the southwestern United States, the typical
time scale for the cutting and filling of large, valley-floor channels is approximately 1000
years (Waters and Haynes, 2001; Mann and Meltzer, 2007). Although the oscillating
45
nature of alluvial channels has been most well studied in arid fluvial systems, they are not
unique to arid fluvial systems. For example, large meandering channels can exhibit
spatial and temporal oscillations in which lateral meandering and deposition occurs
preferentially in relatively low-gradient sections of the valley floor, which then builds a
steep alluvial toe slope that facilitates entrenchment and narrowing downstream (Jones
and Harper, 1998). The non-steady-state nature of channel geometries is also significant
because the stratigraphic record contains only depositional channels within the infill (i.e.,
only the final stage of active-incision is preserved as a scour surface). As such, the
paleoslope estimation method uses an inherently biased proxy for flow depth during
transport of the coarsest load if channel dimensions are measured from an individual
stacked channel within an infilling succession. Given these considerations, it is
reasonable to expect that threshold slopes for entrainment in natural rivers vary
stochastically in time. Within this conceptual model, time periods of lower-than-average
threshold slopes are associated with an incised channel state triggered by climatic
changes or autogenetic dynamics.
If the threshold slope of entrainment varies stochastically through time, then the
transport of coarse sediments such as gravel is no longer controlled by just an average
threshold condition. Instead, gravel may be able to prograde at channel slopes lower than
the minimum value predicted by paleoslope estimation theory, albeit for geologically
brief periods of time when threshold slope is lower than average. In this paper, we
explore the hypothesis that gravel transport is a function of both the mean and coefficient
of variation in threshold slope within a numerical modeling framework. Specifically, we
46
pose the following question: given a sufficient period of time, can gravels prograde
several hundred kilometers from their source regions at slopes significantly lower than
the threshold slope predicted by equation (1) given realistic variations in the threshold
slope of entrainment in space and/or time?
2. Numerical Model Description
We used two types of numerical models in order to test our hypothesis that
gravels can be transported over long distances at regional slopes lower than the threshold
slope predicted by the paleoslope estimation method. The first model, herein called the
isolated sedimentary basin model, represents a gravel wedge prograding along a
horizontal plane in response to a sediment source at a fixed position at the upstream end
of the basin. Deposition and sediment reworking along a channel profile is simulated
using a numerical model that includes mass conservation combined with a slopedependent transport relationship that contains a grain-size-dependent threshold value.
This first model allows us to assess the effects of climate change and autogenetic
processes alone on gravel progradation because sediment supply is held constant and
tectonic movements are not considered. The second model, herein called the
dynamically-coupled postorogenic foreland basin model, represents a gravel wedge
prograding within a foreland basin in response to erosional unloading of an adjacent
postorogenic mountain belt. This model allows us to observe the effects of feedbacks
between erosion in the mountain belt and the evolution of the adjacent sedimentary basin.
The motivation for applying two types of models is that we want to isolate the
effects of stochastic variation in threshold slopes on gravel transport in our isolated
47
sedimentary basin model independent of tectonics and then compare the results to those
of a model that includes tectonics by coupling a basin and its adjacent mountain belt. The
dynamically-coupled postorogenic foreland basin model allows us to determine whether
or not feedbacks between sediment supply to the basin and the base level of erosion for
the mountain belt (controlled by transport efficiency of sediments across the basin) act in
concert to control the rates and slopes at which a gravel wedge can prograde within a
basin. In a numerical modeling study, Paola et al. (1992) showed that varying sediment
supply and accommodation rates sinusoidally through time had significant effects on the
regional slope of an aggrading alluvial fan. The three long-runout gravels described by
Heller et al. (2003) are each associated with a particular period of tectonic activity during
which sediment supply and accommodation were not constant. For this reason, we
include both a dynamically-coupled model of a mountain belt interacting with an adjacent
foreland basin in addition to a model of an isolated basin with a prescribed sediment
supply.
In both the isolated sedimentary basin and dynamically-coupled postorogenic
foreland basin models, threshold slopes for entrainment are randomly sampled from a
lognormal distribution at a regular prescribed interval of time. We use a lognormal
distribution because it is the simplest (i.e. two-parameter) distribution available for a
positive-definite quantity such as channel slope, and also because it provides a good fit to
actual variations observed in modern gravel-bed rivers. Lognormal distributions are
defined by a mean and a coefficient of variation given by:
48
Cv = σSc / S c
(2)
where Cv is the coefficient of variation, σSc is the standard deviation of threshold slope,
and S c is the mean of threshold slope. As the Cv of the threshold slope distribution
increases, the recurrence interval between periods of threshold slopes significantly lower
than average decreases. To estimate an appropriate value for Cv, the mean and standard
deviation of the natural logarithm of the observed threshold slopes, i.e. μ and σ, were
calculated from the Church and Rood (1983) dataset and related to σSc and S c using the
following expressions (Turcotte, 1997):
Sc = e
1
μ + σ2
2
( )
σSc = μ e σ
2 −1 1 / 2
(3)
(4)
The mean and coefficient of variation of the Church and Rood (1983) channel slope data
were calculated using equations (3) and (4) to be 0.013 and 3.045, respectively. A
coefficient of variation greater than 1.0 for a lognormal distribution indicates that the
most frequent slope (mode) is significantly less than the average slope (mean). The most
frequent slope for gravel-bed rivers in the Church and Rood (1983) dataset is an order of
magnitude lower than the average slope and the range of possible slopes spans 3 orders of
magnitude between 10-4 and 10-1. When the slope data are normalized for average flow
depth and median grain size, however, the coefficient of variation in the observed slopes
of modern gravel-bed rivers decreases to 1.66 (Figure 1A). This value for Cv is more
appropriate as an estimate of the variability of threshold slopes of individual gravel-bed
49
rivers through time because mean flow depth and median grain size are far more variable
between rivers (as in the Church and Rood (1983) dataset) than within an individual river
to which the paleoslope estimation method might be applied. A threshold slope of 0.0014
is predicted by equation (1) for a gravel-bed river with an average flow depth of 1.68 m
and a median bed-material grain size of 25 mm. We use this threshold slope as our
representative value in the simulations described below because it is within the range of
threshold slopes calculated for the Ogallala Group, a type-example long-runout gravel
deposit. When a Cv value of 1.66 and a mean threshold slope of 0.0014 are used to
generate a lognormal distribution, approximately 70 percent of threshold slopes are less
than the mean (Figure 1B).
In both types of models we consider here, i.e. in the isolated sedimentary basin
model and in the dynamically-coupled postorogenic foreland basin model, gravel
transport is modeled via a slope-dependent transport relationship with a grain-sizedependent threshold value. Sediment transport is used, together with conservation of
mass (Exner’s equation), to model alluvial erosion and deposition along a 2D channel
longitudinal profile:
q s = k g (S − S c ) ,
S > Sc
qs = 0 ,
S ≤ Sc
[
k g max(S upstream − S c ,0) − max(S downstream − S c ,0)
∂q
∂h
=− s =
∂t
∂x
∂x
(5)
]
(6)
where qs is the volumetric unit sediment flux (m2/s), kg is the transport coefficient
(m2/yr), S is the local channel slope (unitless), h is the channel-bed elevation (m), t is
50
time (yr) and x is distance along the channel from the drainage basin headwaters (m). The
transport coefficient kg can be estimated from an empirical relationship that depends on
discharge and river type (Paola et al., 1992). Paola et al. (1992) estimated transport
coefficients of approximately 1.0 x 104 and 7.0 x 104 (m2/yr) for braided and meandering
rivers, respectively, draining catchments of length equal to 100 kilometers and a mean
annual precipitation of 1 m. We chose a value of 1.0 x 104 (m2/yr) for our numerical
experiments, which corresponds to braided, gravel-bed streams with relatively small
drainage areas.
Threshold slope (Sc) values are sampled from a lognormal distribution with a
mean of 1.4x10-3. The coefficient of variation is varied between experiments but
nominally has a value of 1.66 (i.e., the value calculated for naturally occurring rivers
from the Church and Rood dataset). Once the threshold slope is determined for a given
point along the channel longitudinal profile, that value is held constant for the prescribed
sampling period, which we defined as the length of time that the channel is characterized
by a particular threshold slope. The sampling period value may have a significant effect
on both types of models considered in this paper because it determines whether the
simulated alluvial channels remain in a state of low threshold slope for long periods of
time or whether channels rapidly jump back and forth between states of enhanced or
limited gravel entrainment.
The time scales of autogenetic and allogenic cut-and-fill cycles documented in
natural river systems (e.g. Waters and Haynes, 2001; Mann and Meltzer, 2007; Harvey et
al., 2011) can be used as a guide for choosing sampling periods for both models.
51
Autogenic processes that lead to changes in alluvial channel position through time (i.e.,
such as channel meandering and avulsion) have received much attention in the literature
(Miall, 1996). However, very few studies have focus on autogenic processes that lead to
alluvial channel incision. In the southwestern US, for example, the recurrence interval of
Late Holocene arroyo cut and fill cycles for the San Pedro River that drains a region of
1,000 km2 has been well constrained (Waters and Haynes, 2001). Based on the
radiocarbon dates at the base of each scour, arroyos were incised and filled within 500 to
4,000 years with a predominant period of approximately 1,000 years. Similar trends were
observed for Holocene cut and fill cycles for the Wildhorse Arroyo and Archuleta Creek
of the Upper Cimarron River drainage basin in New Mexico (Mann and Meltzer, 2007).
Radiocarbon and OSL dating of arroyo cycles in Buckskin Wash, UT revealed that at
least 4 cut and fill cycles have occurred since 3 ka, and thus, the predominant period was
between 500 to 1,000 years (Harvey et al., 2011). Although the periodicity of cut and fill
cycles in the southwestern US would be different than that of channels located in other
climates, we infer that 1,000 years is a reasonable value for the threshold slope sampling
period. As such, we chose to hold threshold slopes constant over a period of 1,000 years,
which is equal to the time step for both the isolated sedimentary basin and dynamicallycoupled postorogenic foreland basin models.
Another important issue involving sediment transport within our models is
whether the threshold slope should be uniform or varied along the channel profile during
each sampling period. Observation of alluvial fan and valley floor channels reveal
oscillations or instabilities in channel width and slope that appear to have wavelengths on
52
the order of 100 m to 10 km (Pelletier and DeLong, 2004). Therefore, it’s possible to
have discrete sections of the river profile with lower threshold slopes compared to the
majority of the river profile. If this is the case, then gravel transport should primarily take
place within these localized regions of increased slope and low width-to-depth ratios.
This would suggest that models aimed at understanding the role of autogenetic
oscillations in river systems should use spatially non-autocorrelated threshold slopes
sampled at a relatively small spatial wavelength. However, channels also respond to
climatic and autogenic forcings that trigger incision/aggradation along the entire river.
Such cases would suggest that models aimed at understanding the role of allogenetic
events in river systems should use threshold slope values that vary in time but are highly
spatially autocorrelated along the channel longitudinal profile. An example of such an
allogenetic event would be entrenchment along an entire river profile due to a major
decrease in the ratio of sediment to water from upstream. In this paper we consider both
spatially autocorrelated and spatially non-autocorrelated variations in threshold slope
values. When threshold slope values are spatially non-autocorrelated in the numerical
models, the threshold slope is sampled from a lognormal distribution and held constant
over the sampling period for each alluvial channel pixel representing 1 km. When
changes in threshold slope are spatially autocorrelated, a single threshold slope is
sampled and applied along the entire alluvial channel profile for the given sampling
period.
Our dynamically-coupled postorogenic foreland basin model is a 2D model that
solves for the longitudinal profile of a channel evolving due to bedrock erosion and
53
flexural-isostatic response to erosional unloading of a mountain belt in addition to
sediment transport in an adjacent foreland basin subject to stochastic changes in threshold
slopes of entrainment. A 2D model is sufficient for the purposes of this study because at
least some examples of long-runout gravels (e.g. the Ogallala Group conglomerates) are
effectively 2D systems, i.e. gravels were primarily transported perpendicular to a long,
linear mountain belt. While the model is 2D, it should be emphasized that variations in
threshold slope implicitly include variations in channel geometry in the third dimension,
i.e. channel width. The mountain belt topography at the beginning of each simulation
represents the profile for a high elevation plateau (similar to that of the modern central
Andes or Himalaya).
Bedrock channel erosion is simulated using the empirical stream-power equation
(Howard and Kerby, 1983; Whipple and Tucker, 1999). We do not employ a threshold
for plucking in our model, and thus, bedrock incision occurs as long as the slope is
greater than zero. The following is the stream-power equation applied to the bedrock
portion of the dynamically-coupled postorogenic foreland basin model:
∂h
∂h
= −k e A m
∂t
∂x
n
= − k e A1 / 2 S = −k e xS
(7)
where ke is the bedrock erodibility (yr-1 for n = 1.0 and m = 0.5), A is the drainage basin
area (m2) and m and n are constants that determine the dependence of local erosion on
discharge and channel slope. Evidence from theory and field studies predict that the ratio
of m/n is to be near 0.5 (Whipple and Tucker, 1999). Although the slope exponent n can
range between 0.66 and 2.0 depending on the relationship between slope and stream
54
power (shear stress), we chose to make the stream power linearly proportional to the
slope (n = 1.0 and m = 0.5), consistent with the assumption of many other studies (e.g.
Kirby and Whipple, 2001; Snyder et al., 2000). The range of bedrock erodibility (ke) can
be up to five orders of magnitude (10-2 to 10-7) depending on lithology and the values of
m and n (Stock and Montgomery, 1999). We chose a value of 10-6 yr-1 in order to scale
the erosion rate to reasonable values (1 mm/yr or less) during the early portion of the
simulations when the channel slopes in the bedrock portion of the model are highest.
Alluvial deposition occurs within the bedrock drainage basin when erosion exceeds
transport capacity and therefore, bedrock that is temporarily buried by alluvium does not
erode until the overlying sediment is removed. In this way, the transition between the
bedrock and alluvium is free to migrate laterally in response to the coupled evolution of
the mountain belt-foreland basin system in the model.
Coupling a flexural model to a surface-process model is necessary because
unloading of a mountain belt by bedrock incision should lead to isostatic rebound of the
mountain belt, thus modifying sediment accommodation in the adjacent depositional
basin. The flexural component of our model solves for the displacement of a thin elastic
beam subjected to a spatially-distributed vertical load (Turcotte and Schubert, 1992):
D
∂ 4 w( x )
+ (ρ m − ρ s ) gw( x ) = L( x)
∂x 4
(8)
where w is the deflection of the Earth’s crust (m), D is the flexural rigidity (Nm), ρm is
the density of the mantle (kg/m3), ρs is the density of the mountain crust or foreland
sediment (kg/m3), g is gravity (m/s2) and L(x) is the topographic load (kg/ms2). A Fourier
55
transform method is used to solve this PDE at each time step in the model following
Watts (2001). Appropriate flexural rigidities for a high-elevation plateau can range from
between 2.4x1023 and 3.0x1024 Nm when the Poisson’s ratio is 0.25 and the Young’s
modulus is on the order of 1011 Pa (Stewart and Watts, 1997; Jordan and Watts, 2005).
We chose to apply a flexural parameter of 150 km. The flexural parameter is defined as
the following (Turcotte and Schubert, 1992):
⎡
⎤
4D
α=⎢
⎥
⎣ (ρ m − ρ s )g ⎦
1/ 4
(9)
where α is the flexural parameter (km). Rearranging equation (9) and applying values
from table 1 yields a flexural rigidity of approximately 6.8x1023 Nm, i.e. within the range
of values for mountain belts with high plateaus. In the dynamically-coupled postorogenic
foreland basin model, equation (8) is applied twice, first to determine the initial flexural
profile beneath the topographic load at time zero and second to solve for rock uplift in
response to erosional unloading at each time step. The flexural-isostatic component of the
model was validated using comparison with analytic solutions for a line load. A mountain
belt with an average height of 2.8 km and a width of 550 km triggers a forebulge that is
300-400 m high, a foredeep that is 290 km wide and maximum depth of 3 km when the
rigidity is on the order of 6.0x1023 Nm. An issue that arose during the study is: should the
forebulge be a significant topographic barrier between the foredeep and backbulge
basins? A topographically significant forebulge would prevent gravel progradation out of
the foredeep until the alluvial systems could aggrade to the forebulge spill point. Along
the central Andes, however, there is no topographic expression of the forebulge despite
56
the low elevations observed 200-300 km from the thrust front in Bolivia and Northern
Argentina (i.e. 150-500 m a.s.l.) (Horton and DeCelles, 1997). Conversely, the predicted
forebulge region based on 2D flexural modeling in the Himalayan foreland basin is
elevated approximately 300-400 meters above the Ganges plain (Bilham et al., 2003).
Interpreting this relief as exclusively the result of forebulge uplift is complicated by the
presence of pre-existing topography in the Central Indian Plateau. Although the
Himalayan foreland may be an exception, topographically expressed forebulges are not
commonly observed within modern terrestrial foreland basins, most likely due to erosion
and/or dynamic subsidence (DeCelles and Giles, 1996; Catuneanu et al., 2000). In order
for the model to predict a forebulge amplitude that is consistent with modern analogs (i.e.
a maximum of 150-300 m a.s.l.), we beveled the forebulge crest at the beginning of the
model down to the elevation of the initial fluvial profile. However, this initial
topographic adjustment does not modify the flexural response of the forebulge, which is
an important control for sediment accommodation and depositional basin regional slope,
to postorogenic denudation of the adjacent mountain belt.
Parameters for both the isolated sedimentary basin and dynamically-coupled
postorogenic foreland basin models are reported in table 1. The isolated sedimentary
basin model simulates 10 million years of gravel progradation using a time step of 1 kyr
using a pixel spacing of 1 km. The mean threshold slope is held constant at a value of
0.0014 and the threshold slope sampled from the lognormal distribution is applied to the
entire channel profile. Simulations with a range of Cv values and a range of ratios of the
sediment supply to the transport coefficient, qs/kg, were conducted to test the sensitivity
57
of gravel progradation to variations in these parameters. Sediment flux into the basin is
held constant at a rate of 6 m2/yr for the simulations with variable Cv, consistent with a
qs/kg value of approximately 0.00075. We chose this value for sediment supply because it
is equivalent to the average sediment supply coming out of the drainage basin during the
period of highest sediment flux for each of the dynamically-coupled postorogenic
foreland basin model simulations discussed in the Results section. Simulations for each
Cv value are repeated 20 times and the reported results are an average of those model
runs. Following the simulations with variable Cv and qs/kg, we also tested the sensitivity
of gravel progradation rates and slopes to the presence or absence of spatial
autocorrelation in threshold slope along the channel profile.
The dynamically-coupled postorogenic foreland basin model simulates 20 million
years of gravel progradation using a time step of 1 kyr and pixel size of 10 km. The initial
topographic profile for the bedrock drainage basin is an actively uplifting mountain belt
with frontal slopes of 0.011 and a low-relief plateau interior. In the first simulation of this
model type, threshold slopes are sampled from a lognormal distribution with a Cv of 1.66
and a mean value of 0.0014, i.e. equivalent to that of the Church and Rood (1983) dataset
and therefore, representative of natural variations of slopes of channels with similar mean
flow depths and grain sizes. Changes in threshold slope are spatially autocorrelated along
the channel profile. Simulations with spatially non-autocorrelated changes in threshold
slope are not considered here due to the results for comparing the isolated sedimentary
basin model simulations with uniform or non-uniform changes in threshold slopes which
we address in the Discussion section. Topographic surfaces are sampled for display at
58
0.0, 0.25, 1.0 and 20.0 Myr of simulation. We also report on additional simulations that
were ran to test the effect of varying Cv and the sampling period on the evolution of
postorogenic gravel progradation.
3. Numerical Modeling Results
Time-series results for regional slope and the position of the gravel front
generated by the isolated sedimentary basin model with spatially autocorrelated changes
in threshold slope are shown in Figure 2. Regional slopes (defined herein as the ratio of
the relief of the depositional basin to the length of the basin) decrease systematically with
increasing Cv. When the Cv value is greater than or equal to 1.0, the gravel wedge
progrades at regional slopes that are less than the mean threshold slope (i.e., 0.0014).
Thus, the model results suggest that natural rivers with a Cv of 1.66 and the channel
conditions considered here can prograde at regional slopes significantly lower than that
predicted by the paleoslope estimation method.
The location of the gravel front increases as a power-law function of time with an
exponent that increases from approximately 0.5 to 0.8 as the Cv increases from 1 to 8
(Figure 2B). Even prior to 1.0 Myr, clastic-wedge progradation in the model is rapid and
gravels begin to reach downstream distances that are considered “long-runout,” i.e.
greater than tens of kilometers (Heller et al., 2003), when the Cv is high. Beyond 1 Myr,
progradation rates decrease to long term average rates of between 30-55 km/Myr with the
precise value dependent on Cv. Increasing the Cv value by a factor of 8 increases the total
progradation by less than a factor of 2 after 10 Myr. In such cases, gravel-bed channels
with a Cv value comparable to natural rivers, i.e. 1.66, are able to transport gravel up to
59
approximately 400 km away from their source regions given sufficient geologic time (i.e.
10 Myr).
Numerical modeling studies have shown that alluvial basin slopes are also
affected by sediment supply rates and sediment transport rates (Paola et al., 1992).
Therefore, we conducted simulations to determine the variation in the behavior of this
model with respect to variations in qs/kg (Figure 3). We chose not to test the sensitivity of
gravel progradation to qs and kg individually because the regional slope of a clastic wedge
is sensitive to the ratio between these two parameters rather than their individual values.
The regional slopes shown in Figure 3 are the long-term regional slopes illustrated in
Figure 2. When the Cv value is equal to 0, the resulting regional slopes are always greater
than the mean threshold slope (i.e., 0.0014) calculated using equation (1). When
threshold slopes are allowed to vary stochastically in time, long-term regional slopes
decrease below the mean threshold slopes when qs/kg is less than 0.00125. Values of qs/kg
above approximately 0.00125 correspond to cases in which sediment supply to the basin
exceeds the ability of the gravel-bed channels to convey that sediment at the mean
threshold slope. As the qs/kg ratio increases above 0.01, the long-term regional slope
increases as a power-law of qs/kg. In such cases, the gravel channels aggrade with little
dependence on the value of Cv. Conversely, values of qs/kg less than approximately
0.00125 correspond to cases in which the transport capacity at the calculated mean
threshold slope approaches the sediment supply. The qs/kg value used in the analysis
shown in Figure 2 is located in this region of the plot. In this situation the Cv value has a
significant effect on the long-term regional slope. The relationship between regional
60
slope and qs/kg implies that sedimentary basins located adjacent to actively uplifting
bedrock drainages may not be able to produce long-runout gravels with low regional
slopes early on while sediment supplies are high. However, once active uplifting ends the
regional slopes of the gravel-bed rivers can lower in response to infrequent periods of
gravel transport.
Simulations with spatially autocorrelated and non-autocorrelated changes in
threshold slope were conducted to test the sensitivity of gravel progradation to the
presence/absence of spatial autocorrelations in threshold slope along the channel profile.
We found that the sedimentary basin regional slopes were identical in the two cases, i.e.
the results for spatially non-autocorrelated threshold slopes lie directly on top of the
results obtained with spatially autocorrelated changes illustrated in Figure 3. In the case
of spatially-autocorrelated threshold slopes, there are time periods of high effective
transport rates (when threshold slopes are low across the entire basin). One might expect
such cases to have higher effective transport rates compared to cases with spatially nonautocorrelated threshold slopes, but the results of our model do not support this
expectation. In the case of spatially non-autocorrelated variations in threshold slope,
gravel is transported in small episodic steps that are localized in space but occur
commonly through geologic time. In the case of spatially autocorrelated variations in
threshold slope, gravel is transported long distances in brief periods of unusually-low
threshold slope, but then the basin experiences long periods with no transport anywhere
until the entire basin can again achieve a transport condition characterized by a low
threshold slope (e.g. a phase of active entrenchment). Therefore, the selection of either
61
spatially autocorrelated or non-autocorrelated changes in threshold slope does not
significantly affect the results of either the isolated sedimentary basin or dynamicallycoupled postorogenic foreland basin model, provided that sampling periods are small, i.e.
on the order of millennia, compared to the duration of the model.
In addition to climatically or autogenically driven changes in threshold slope,
tectonically driven changes in sediment supply and basin accommodation might affect
both the regional slope and rates at which gravels are transported. Three types of results
are reported for the dynamically-coupled postorogenic foreland basin model: (1) a
description of how bedrock drainage relief, rock uplift rates, sediment supply rates and
regional slope evolve during a simulation with a Cv value that is representative of natural
rivers (i.e., 1.66); (2) a comparison of regional slope results for simulations with varying
Cv values and (3) comparison of regional slope results for simulations with varying
sampling periods.
The first 250 kyr of the simulation with a Cv value of 1.66 is marked by
knickpoint retreat toward the plateau divide and an increase in basin average erosion rates
(Figure 4A). Steep slopes and large contributing drainage areas (i.e. a proxy for
discharge) lead to peak bedrock incision rates of 0.7 mm/yr near the mountain front. As
such, high erosion rates near the mountain front begin to decrease slopes, and thus,
erosion rates over the first 250 kyr as the main bedrock channel adjusts to slower (i.e.,
postorogenic) rock uplift rates. Overall, erosional adjustment of the drainage network
over this time period has not significantly changed maximum erosion rates because the
62
topography after 250 kyr of simulation still closely resembles the original topography at
the end of active deformation.
High basin-averaged erosion rates over the first 250 kyr following the cessation of
active deformation leads to isostatic rebound, gravel input into the depositional basin and
gravel front progradation. Erosional unloading of the mountain belt causes between 40 to
80 meters of rock uplift within the bedrock drainage basin over the first 250 kyr. Bedrock
incision near the topographic divide is an order of magnitude less than incision at the
mountain front in this time period, and as a result peak elevations increase by 40 m at the
topographic divide. Overall, however, the mean elevation of the mountain belt is less than
its original value, as it must be because only passive (isostatic) uplift is occurring.
Isostatic rebound in the adjacent depositional basin increases proximal alluvial slopes
because rock uplift exponentially decreases away from the mountain front. High basin
average erosion rates during the first 250 kyr also lead to an increase in gravel flux into
the depositional basin. Gravel flux reaches a maximum value of 10 m2/yr and an average
of approximately 5-6 m2/yr at this time. As a consequence of increasing sediment supply
and proximal channel slopes, the gravel front rapidly progrades approximately 150 km
(i.e., a time averaged rate of 0.6 m/yr) from the mountain front in the first 250 kyr.
After 1 Myr of postorogenic time, basin-averaged erosion rates continue to
decrease as the increase in peak elevations and bedrock channel slopes near the
topographic divide is accompanied by a decrease in channel slopes near the mountain
front (Figure 4A). In contrast to the initial 250 kyr, the highest erosion rates (0.6 mm/yr)
are concentrated within the intermediate portion of the drainage basin instead of at the
63
mountain front. Reduced slopes and temporary sediment storage on the timescale of one
hundred thousand years both act to limit erosion rates within the lower portion of the
drainage system. Overall, the basin-averaged erosion rates at this point in time approach
rates expected for postorogenic denudation (Matmon et al., 2003; Reiners et al., 2003).
Erosion in the drainage basin leads to approximately 100 m and 140 m of surface uplift at
the mountain front and topographic divide over the 0.75 Myr period. Isostatic rebound in
the mountain belt is accompanied by subsidence in the forebulge region. As the surface
topography lowers in the forebulge region and sediment flux remains high, the distal toe
of gravel deposition migrates 100 km further from the mountain front.
Following 20 Myr of postorogenic denudation, basin-averaged erosion rates are
on the order of 0.01 (mm/yr) as bedrock incision is concentrated near the topographic
divide where steep slopes still persist (Figure 4A). Both peak elevation and erosion rates
are comparable to values characteristic for postorogenic mountain belts that have only
been eroding for tens of millions of years. Approximately 610 and 3780 m of exhumation
occurs at the mountain front and topographic divide respectively over the 19 Myr period
between 1 Myr and 20 Myr following the start of the simulation. Isostatic rebound at the
mountain front due to erosional unloading of the mountain belt continues to steepen
proximal foreland slopes. The distal toe of gravel deposition migrates an additional 300
km into the foreland basin in this time interval. Gravel progradation persists despite the
factor of 3 decrease in gravel flux from the bedrock drainage system compared to the
0.25-1.0 Myr time interval (Figure 4B).
64
Changes in rock uplift and sediment supply rates have a significant effect on
regional slope of the foreland basin. Figure 4B shows the time-dependent behavior of the
regional slope. The most rapid change in regional slope (i.e. decreasing from values of
8.0x10-4 to values of approximately 5.0x10-4) occurs early in the simulation (prior to 200
kyr). Initially, increasing gravel flux from the bedrock drainage and non-uniform rock
uplift rates near the mountain front lead to rapid gravel migration into the foredeep. This
initial period is followed by a regional slope increase that lasts until approximately 1.5
Myr when the toe of the gravel wedge reached the distal foredeep and mean gravel flux
from the mountain front begins to decrease. A topographically significant forebulge
would impede gravel progradation beyond the foredeep as the distal toe is forced to
aggrade to a forebulge spill point. In this case, however, the forebulge is eroded by the
fluvial system to be in grade with the foredeep channel. Beyond 1.5 Myr, the gravel
wedge toe is located in the forebulge to backbulge region where rock uplift rates are
extremely low, and thus, the decrease in regional slope with time is predominantly
controlled by decreasing gravel fluxes at the mountain front and the stochastic variation
in threshold slopes.
Regional slopes are nearly linear in log-log space beyond 8-9 Myr of
progradation, and thus represent a power-law relationship between regional slope and
time (Figure 4B). It can also be argued that the decrease in regional slope is nearly linear
as early as 2 Myr following the end of active uplift. When power-law curves are fit to the
data in Figure 4B through linear regression of the log-transformed data between 2 and 8
Myr and between 8 and 20 Myr, the resulting power-law exponents are -0.11 and -0.34
65
respectively (with correlation coefficients of approximately -0.982 and -0.999
respectively). The differences in exponent values may be related to changes in sediment
supply and foreland slopes near the toe of the gravel wedge.
Stochastic changes in threshold slope are the main mechanism for driving gravel
progradation below the mean threshold slope, and thus have a significant effect on the
long term decrease in the foreland basin regional slope with time. The variation in
postorogenic regional slope as a function of the coefficient of variation of threshold slope
(i.e., ranging between 0.0 and 3.0) and time is summarized in Figure 5. Heller et al.
(2003) proposed that the gravel wedge must achieve and maintain a mean threshold slope
to prograde gravel. Figures 4 and 5, however, show a decrease in regional slopes through
time that suggests that gravel-dominated foreland basins do not maintain the predicted
mean threshold slope (i.e., 0.0014 for our models) as time progresses. When there is no
variation in threshold slope the long-term regional slope approaches the mean threshold
slope predicted by equation (1) for the given parameters. However, the regional slope
may not decrease to the mean threshold slope until the sediment input from the drainage
basin approaches the transport capacity as demonstrated in Figure 3. When the Cv is 0.5,
approximately 5.5 Myr of postorogenic gravel transport is required to reduce regional
slopes below the calculated mean threshold slope value. Regional slopes never increased
above the calculated mean threshold slope when the Cv is greater or equal to 1.0, and
thus, the results suggest that the rate of gravel progradation for a given grain size far out
into the foreland basin is strongly dependent on Cv and time in addition to the value of
the transport coefficient. When power-law curves are fit to the data a negative correlation
66
exists between the Cv value and the power-law exponent for slope decrease after 7 Myr.
The relationship between the Cv and power-law exponent is best approximated by a
logarithmic function (correlation coefficient value of 0.9869). The negative relationship
between Cv and the power-law exponent occurs because higher variability in threshold
slopes increases the transport capacity of gravel-bed channels early on when sediment
fluxes are high (i.e., between 0.25 and 1 Myr) and prevents significant aggradation. As
such, regional slopes are already low at the beginning of the final phase of regional slope
decrease.
Thus far we have shown how regional slopes respond to changes in the threshold
slopes held constant over sampling periods of 1,000 years. However, gravel-bed channels
that maintain threshold slopes over longer timescales may evolve differently than
channels that vary threshold slopes over shorter timescales. Simulations using the
dynamically-coupled postorogenic foreland basin model were run for different sampling
periods to determine the dependence of gravel progradation on the sampling period
duration (Figure 6). We define sampling period as the length of time over which the
threshold slope for gravel transport is held constant before a new threshold slope is
picked from the lognormal distribution. The curves illustrated in Figure 6 represent an
average of trials conducted for sampling periods of 500 yr, 1 kyr, 10 kyr and 100 kyr.
The solid line in Figure 6 represents the sampling period applied to the simulation
with a Cv of 1.66. The decrease in long-term regional slope (i.e., the trend in regional
slope beyond 10 Myr) averaged over many trials remains unchanged as the sampling
period is decreased by a factor of two or increased by one to two orders of magnitude. By
67
10 Myr, each of the sampling period simulations have experienced enough changes in
threshold slope that the regional slope values resulting from the different sampling
intervals converge. At shorter timescales, regional slopes for simulations with 10-100 kyr
sampling periods are greater by a factor of two than the slopes for simulations with 0.5-1
kyr sampling periods. However, this trend may disappear when regional slope is averaged
over more trials. Overall, the first order trend in regional slope, i.e. an increase in regional
slope between 0.3 and 1 Myr followed by a long-term decrease below mean threshold
slope, occurs in each simulation. As such, the first order behavior of the results for either
model is not significantly biased by the size of the sampling period.
4. Discussion
Results from the simulations presented here suggest that allowing the threshold
slope of gravel entrainment to vary stochastically through time in a lognormal fashion has
a significant effect on the regional slopes at which gravels prograde in a depositional
basin. When gravel progrades as a simple wedge with a constant sediment supply rate in
the absence of tectonic influences, the regional slope reaches a constant value after only
approximately 1 Myr that is dependent on Cv and qs/kg. According to our model results,
gravel-bed channels that experience greater variation in threshold slope values develop
lower regional slopes than those with relatively constant threshold slope values through
time. This behavior is independent of whether the changes in threshold slope are
localized (i.e., autogenically forced instabilities in the channel cross-section) or
autocorrelated along the entire river profile (i.e., climatically forced entrenchment). If the
qs/kg are sufficiently low and/or the Cv values are sufficiently high the regional slope can
68
be lower than the threshold slope calculated with the paleoslope estimation method
during gravel transport. During postorogenic denudation of a coupled orogensedimentary basin system, the regional slope of a gravel-dominated basin decreases in a
power-law relationship with time following an initial adjustment period of approximately
1 Myr. The timescale of the adjustment period depends on how rapidly the sediment
supply begins to decrease following the end of active-uplift. Regional slope decrease
initially occurs because channels are adjusted to higher sediment supply created by an
actively uplifting orogen. However, the long-term regional decrease results from a
combination of a decrease in sediment supply as the orogen loses relief and a stochastic
variation in threshold slope. The duration and magnitude of the long-term regional slope
decrease is dependent on the Cv, kg and the initial relief of the orogen. A power-law
relationship with time predicts that the postorogenic foreland basin regional slope can
become much lower than the mean threshold slope predicted by paleoslope-estimation
theory if the qs/kg ratio is sufficiently small. In the absence of major axial rivers, preexisting topography and additional tectonic events that affect the postorogenic foreland
basin, gravel units will continue to prograde away from the mountain belt through time
over a period of more than 20 Myr.
The long-term regional slope of a sedimentary basin and rate of decrease in slope
through time depend on the Cv value. Differences in the frequency of periods of lowerthan-average threshold slopes as a function of Cv and time lead to the dependence of
regional slope on the value of Cv. Simulations with larger coefficients of variation should
experience a shorter amount of time between gravel-transporting intervals that can
69
transport gravel in the distal portions of the basin, which leads to an increasingly negative
exponent value in the power-law relationship between regional slope decrease and time.
This behavior has been described in the statistics literature as a “threshold-crossing”
problem. For example, the probability of a threshold slope on the order of 1.0x10-4 is
extremely low for a coefficient of variation of 1.0 (less than 4 percent frequency). When
sediment supply rates and foreland basin slopes are highest during the early stages of
postorogenic rebound, the time periods between gravel-transporting intervals is relatively
short. As gravel is transported further into the sedimentary basin during periods of lower
threshold slope, alluvial channel slopes decrease. Therefore, the duration between graveltransporting intervals (i.e. those with sufficiently low threshold slopes) at the distal toe of
the gravel wedge continually increases. Increasing the coefficient of variation to 3.0,
increases the probability of threshold slopes on the order of 1.0x10-4 (approximately 16
percent frequency). Although these probabilities seem small, the number of millennia out
of a 20 million year simulation where the threshold slope is on the order of 10-4 can be
quite large (102 to 103 millennia).
The results for both models suggest that gravels can prograde hundreds of km at
low regional slopes over a few million years when sample periods are on the order of
millennia. Although 1,000 years is an appropriate (mid-ranged) value for the sampling
period, the timescales of cut and fill cycles can vary between 500 to 4,000 years. This
motivates the question: as channels spend longer periods of time at a given threshold
slope (i.e., longer sampling periods), must the long term progradation of gravels happen
more rapidly or at significantly lower slopes than channels that more frequently change
70
their threshold slopes? We infer from our model results that the long-term progradation
rates (> 10 Myr) and foreland behavior are approximately time invariant for the tested
sampling periods (Figure 6). As long as the duration of the gravel transport is much larger
than the sampling period, then the long term gravel progradation will occur at similar
rates and regional slopes. At shorter timescales, autogenic or climatic processes that lead
to changes in flow depth, and thus threshold slopes, every 0.1-1 kyr are more efficient at
prograding gravel than changes in flow depth on timescales of 10-100 kyr when the Cv is
representative of modern rivers.
If variations in channel width-to-depth ratio over geologically brief timescales
(i.e., 0.001 to 1.0 Myr) occurred during gravel transport, then the internal stratigraphic
architecture of the gravel deposits should reflect this behavior. The Buckhorn
Conglomerate member of the Lower Cretaceous Cedar Mountain Formation located in
southern Utah and the Upper Conglomerate of the Lower Cretaceous Cloverly Formation
of central Wyoming, for example, are composed of coalesced lenticular conglomerate
bodies that contain planar to trough cross-stratification (DeCelles and Burden, 1992;
Currie, 1998). Erosional surfaces that mark a cycle of cutting have not been identified
within the coalesced gavel bodies; however, such surfaces may be difficult to detect
within continuous conglomerate successions (Miall, 1996). A lack of developed flood
plain units within or laterally correlative to the conglomerates and the lateral continuity
of gravelly to coarse sandy deposits lead to the interpretation that the gravel bodies
represent deposition in a series of unstable, interconnected, braided fluvial channels that
reworked the entire width of paleovalleys in which the gravels were deposited. Due to the
71
lateral instability of braided channels and low accommodation rates during deposition it
is not surprising that soils and fine grained floodplain deposits were not preserved in the
valley fill, and thus, evidence of periods of channel stability or entrenchment over
geologically-brief periods of time may only be preserved in the unconformity surfaces
adjacent to the paleovalleys. Evidence for cycles of channel incision and paleosol
formation during the deposition of conglomerates is seen within the basal Shinarump
conglomerate of the Upper Triassic Chinle Formation (Blakey and Gubitosa, 1984;
Hasiotis et al., 2000). Gravelly to sandy bodies within the Shinarump conglomerate
paleovalley fill were also interpreted to be deposited by an unstable, braided river system
due to the lack of fine grained, horizontally laminated floodplain deposits and the
abundance of trough-cross stratified dunes and planar cross-stratified bars (Blakey and
Gubitosa, 1984). Significant scour surfaces between individual channel stories with relief
ranging between 1-2 meters are observed within outcrops located in northern Arizona and
southern Utah. However, its unclear if this magnitude of incision was accomplished over
a significant period of time or during a single flood event. Ephemeral alluvial rivers with
drainage basin areas on the order of 50-100 km2 have been observed to incise their
channel beds up to 2 m during floods where average flow depths were between 0.3 to 2m
(Hassan et al., 1999; Hooke and Mant, 2000). In the Petrified Forest National Park of
Northern Arizona, multiple cycles of valley cut-and-fill have been identified in the lower
part of the Chinle Formation (Hasiotis et al. 2000). Lenticular channel deposits of the
Shinarump member cut into contemporaneous overbank deposits that show slight to
moderate pedogenesis.
72
Clear evidence for cycles of entrenchment and periods of low deposition adjacent
to the active channel can be found within the internal stratigraphic architecture of the
Tertiary Ogallala Group observed in Nebraska. Valley fill deposits of the Ogallala Group
are separated by major erosional surfaces and contain Miocene fossil assemblages of
distinctly different ages, which suggest multiple cycles of cut and fill occurred after the
initial incision of the main paleovalleys (Diffendal, 1982). In some outcrops the younger
deposits cut completely through to basal Ogallala and into the underlying mid-Tertiary
stratigraphy. Further south in Texas and New Mexico, cycles of deposition and
pedogenesis can be seen within the fine sand to coarse silt lithofacies of the Ogallala
Group. In some outcrops eolian deposits that contain cycles of deposition and
pedogenesis cap fluvial sections in the paleovalley fill and paleo-uplands (Gustavson and
Holliday, 1999).
Cycles of entrenchment and fill, that cause periods of low threshold slopes over
millennia timescales, have been directly observed and inferred from Pleistocene and
Holocene alluvial fan and valley floor river outcrops. Although climatic conditions for
the Quaternary may be quite different from other periods of time, cutting and filling
cycles are not limited to the Quaternary. For example, DeCelles et al. (1991) interpreted a
series of 5th order lithosomes and 5th order surfaces (as defined by Miall, 1995) within
alluvial fan deposits from the Paleocene Beartooth Conglomerate in Montana as a cycle
of entrenchment and backfilling. Holbrook (2001) observed a continuum of channel-form
bounding surfaces within the Cretaceous Muddy Sandstone of southeastern Colorado that
represent river scour at different scales. Nested valleys or bundles of channel belts
73
truncated and bound by higher order surfaces were present throughout the formation.
These nested valleys were interpreted to analogous to modern cut and fill deposits that
form over timescales of 103 years. Based on the observation of the Western US longrunout gravels and other evidence for entrenchment and fill cycles contained within the
rock record, invoking stochastic variations in threshold slope to drive gravel transport at
low gradients over long distances is realistic.
Our modeling results suggest that ancient orogens that have undergone
postorogenic denudation under ideal conditions (e.g. sufficient relief and appropriate
drainage lithologies for generating gravel) should produce long-runout gravels if given
sufficient time. However, it’s unclear if long-runout gravels have not been described for
other ancient orogens outside of the western U.S. because they were not recognized as
having been transported considerable distances from their source region or the conditions
for producing long-runout gravels are rare. One explanation for why long-runout gravels
might be rare is that the ability of gravel clasts to resist grain size reduction through
abrasion may be an important factor (Kodama, 1994; Lewin and Brewer, 2002). As clasts
saltate along the channel bed or are at rest, collisions with the bed or other clasts cause
the grain size to decrease by shedding off smaller particles. Grain size reduction generally
follows a power-law with distance from the source of the clasts. Although channel
conditions may be suitable for transporting gravel out to the distal foreland basin, the
gravel may be broken down into sand sized grains within 50-100 km from where they
were sourced depending on the lithology. Interestingly, the Shinarump and Lower
Cretaceous gravels are dominated by resistive lithologies. For example, approximately 24
74
and 72 percent of point counts within samples from the Buckhorn Conglomerate were
composed of chert nodules and monocrystalline quartz (Currie, 1998). Chert and
quartzite are also prevalent within the Lakota, Cloverly and Shinarump conglomerates
(Blakey and Gubitosa, 1984; Zaleha et al., 2001). We infer from the lithologic trends of
Lower Cretaceous and Shinarump conglomerates that gravels require lithologies that are
resistant to grain size reduction in order to be transported 100s of kilometers from their
source regions; although, the Ogallala Group conglomerates are an exception.
Another possible explanation for why ancient orogens do not commonly produce
long-runout gravels is due to the lithology of the eroding drainages. Toward the eastern
edge of the central Andes, for example, Tertiary synorogenic sediments are being
exhuming within the Subandean Zone, which is the active fold and thrust belt today.
Within the Tertiary units, the Yecua and Tariquia Formations that make up
approximately half the thickness of the Tertiary units, mainly compose of sandstones and
mudstones with some conglomerate units present (Uba et al., 2005). Consequently,
studies of the large megafans that drain this region in southern Bolivia and northern
Argentina observed limited gravel to abundant sand and mud along the entire megafan
profile (Horton and DeCelles, 2001). Conglomerates should be absent from the rock
record if material that is prone to erode into finer grain sizes is being exhumed in the
bedrock drainage regions that are the predominant supply of sediment to the depositional
basin.
5. Conclusions
75
The evolution of gravel transport within foreland basins is significantly influenced
by the stochastic variation in threshold slopes. When the gravel supply is held constant
and subsidence is negligible, the regional slope of a prograding gravel wedge reaches a
long-term value that is dependent on the values qs/kg and both the mean and coefficient of
variation of the threshold slope of entrainment. When the Cv value is less than one, the
effective regional slope remains equal to or greater than the mean threshold slope and
gravel progradation is minimized. However, when the Cv value is greater than one and
the basin sediment supply is low, gravel can be transported at regional slopes that are
well beneath threshold slopes predicted by the paleoslope estimation method. As such,
caution should be exercised when evaluating regional slopes measured from the rock
record in a tectonic context.
The results of our dynamically-coupled postorogenic foreland basin model show
that the decrease of foreland basin regional slopes closely follows a power-law
relationship with time, although variability in sediment supply and rock uplift cause
significant scatter in foreland basin slopes through time. This implies that gravel will
continue to prograde in the foreland basin following the end of active uplift for a
significant period of time. The power-law regional slope decrease with time observed in
the model does not appear to depend on the sampling period beyond 10 Myr as long as
the period does not approach a million years. The spatial autocorrelation of threshold
slope change also has little effect on the first order behavior of regional slope decrease,
and thus, either climatic or autogenic forcings should be sufficient to prograde gravel at
extremely low regional slopes.
76
Our model results suggest that under ideal conditions, long-runout gravels can be
formed within a period that is on the order of a few million years if threshold slopes are
variable through time. The three described long-runout gravels of the western US contain
lithologies that are highly resistive to grain size reduction through abrasion which we
infer to also be a necessary condition for the transport of gravel long distances. Both
macroscale and internal geometries within the Lower Cretaceous and Ogallala gravels
support the possibility of significant variation in channel width-to-depth ratios through
time. Therefore, the Shinarump, Lower Cretaceous and Ogallala gravels need not have
required a tectonically driven increase in slope above the observed slopes today to
transport large distances from where they were sourced.
Acknowledgements
This research was supported in part by ExxonMobil funding. We thank Doug
Jerolmack for helpful conversations.
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Tables, Figures and Figure Captions:
TABLE 1. PARAMETERS FOR MODEL SIMULATIONS
Parameter
Isolated
Sedimentary Basin
Model
Dynamically-Coupled
Postorogenic Foreland
Basin Model
1.0x107
2.0x107
ρm (kg/m3)
NA
3300
ρc (kg/m3)
NA
2750
ke (yr-1)
NA
1.0x10-6
kg (m2/yr)
1.0x104
1.0x104
S c (unitless)
0.0014
0.0014
Model Duration (yr)
Figure 1: Channel slope statistics applied to calibrate the variation in threshold slope. (A)
Cumulative frequency distribution for normalized slopes determined from slope data
collected by Church and Rood (1983) for gravel-dominated rivers in North America. (B)
Cumulative frequency distribution for model threshold slopes. Vertical, dotted line
87
represents the location of the threshold slope that is predicted by paleoslope estimation
theory.
Figure 2: Isolated sedimentary basin model results. (A) Average regional slopes and (B)
gravel front positions through time for gravel-bed channels with different Cv values.
Figure 3: Relationship between the regional slope of a prograding clastic wedge in an
isolated sedimentary basin and the ratio of sediment supply to transport coefficient for
88
various values of Cv. The solid curves represent simulations with correlated changes in
threshold slope along the entire channel profile. The dashed horizontal line represents the
mean threshold slope of 0.0014 and the dashed vertical line represents the sediment
supply to transport coefficient ratio for simulations shown in Figure 2.
Figure 4: Dynamically-coupled postorogenic foreland basin model results for the
simulation with a Cv value of 1.66. (A) Topographic cross sections and (B) sediment flux
and regional slope time-series following a cessation of active uplift. The solid, dotted,
dashed and dashed-dotted dark lines in Figure 4A represent snap-shots in time of
topography at intervals of 0, 250 kyr, 1 Myr and 20 Myr following the cessation of
tectonically driven rock uplift. The vertical tick marks represent the downstream limit of
gravel deposition during those instants in time. In Figure 4B the regional slope and
sediment influx rate are represented by the solid and dotted lines respectively.
89
Figure 5: A summary of the dynamically-coupled postorogenic foreland basin model
simulations for the coefficient of variation of threshold slope. The mean threshold slope
calculated using equation (1) for the assumed channel conditions is represented by the
dashed line. Each sensitivity study curve is an average of individual trials for a given Cv
value.
90
Figure 6: Resulting regional slopes for a range of sampling intervals for the dynamicallycoupled postorogenic foreland basin model. The dot-dashed, solid, dashed and dotted
lines represent 500 yr, 1 kyr, 10 kyr and 100 kyr sampling intervals respectively.
91
APPENDIX B: SIMULATING FORELAND BASIN RESPONSE TO
MOUNTAIN BELT KINEMATICS AND CLIMATE CHANGE FOR
THE CENTRAL ANDES: A NUMERICAL ANALYSIS OF THE
CHACO FORELAND IN SOUTHERN BOLIVIA
Manuscript for submission to Tectonics
Todd M. Engelder a and Jon D. Pelletier
Department of Geosciences, University of Arizona, 1040 E. Fourth St., Tucson AZ,
85721, USA,
a
Corresponding Author, email: engelder@email.arizona.edu, fax: (520) 621-2672
Abstract
The role of crustal thickening, lithospheric delamination, and climate change in driving
surface uplift in the central Andes in southern Bolivia and changes in the creation of
accommodation space and depositional facies in the adjacent foreland basin has been a
topic of debate over the last decade. Interpretation of structural, geochemical, geomorphic
and geobiologic field data collected from the region has led to two proposed end-member
Tertiary surface uplift scenarios. A “gradual uplift” model proposes that the rate of
surface uplift has been relatively steady since deformation propagated into the Eastern
Cordillera during the Late Eocene (McQuarrie et al., 2005). In this scenario, the mean
elevation of the region was > 2 km a.m.s.l. by the Late Miocene or earlier. Alternatively,
a “rapid uplift” model suggests that the mean elevation of the Altiplano was < 1 km
a.m.s.l. and that the peaks of the Eastern Cordillera were more than 2 km below their
modern elevations until 10 Ma (Garzione et al., 2008). Determining which of these uplift
scenarios is most consistent with the stratigraphic record is complicated by global climate
changes and lithospheric delamination during this period. In this study, we use a coupled
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mountain belt/sediment transport model to predict the foreland basin stratigraphic
response to end-member surface uplift scenarios, lithospheric delamination and climate
change. Our model results indicate that the location and height of the migrating
deformation front plays the dominant role in controlling changes in accommodation
space and grain size within the foreland basin. Changes in accommodation and sediment
supply rates related to climate change and lithospheric delamination events appear to be
secondary to crustal thickening and propagation of the deformation front. Based on
stratigraphic data, the Eastern Cordillera likely gained significant elevation prior to 10
Ma, which is in contrast with recent proposals that most of the modern elevation was
obtained during the Late Miocene.
Keywords- sediment transport, foreland basin, central Andes, surface uplift,
delamination, climate change
93
1. Introduction
Sediments eroded from mountain belts are primarily deposited in foreland basins,
which are elongate troughs created by flexural loading of the lithosphere adjacent to
mountain belts (DeCelles and Giles, 1996). Foreland-basin stratigraphy is recognized to
be a useful dataset for reconstructing climate and crustal deformation histories within
ancient mountain belts (e.g., Heller et al., 1988; DeCelles et al., 1998; Marzo and Steel,
2000, Uba et al., 2007). The fluvial systems that transport the sediment from mountain
belts are sensitive to changes in sediment supply, discharge and sediment accommodation
rates, and thus, the foreland-basin stratigraphy deposited by these systems is a partial
record of changes in climate and the geometry of the mountain belt load through time.
The flexural profile of foreland accommodation depends on the rigidity of the lithosphere
that is underthrusted beneath the mountain belt as well as the width and elevation of the
mountain belt load. If the foreland basin geometry and rigidity of the underthrusted
lithosphere are constrained, then it is possible to infer changes in the mountain belt load
that occurred during the development of the foreland basin. As such, numerous studies
have inferred tectonic histories for ancient mountain belts by fitting foreland basin
geometries predicted by a coupled mountain belt and foreland basin numerical model to
observed foreland basin isopach data (e.g., Toth et al., 1996; Ford et al., 1999; GarciaCastellanos et al., 2002; Prezzi et al., 2009).
The eastern margin of the central Andes (i.e., the portion of the mountain belt that
is to the east of the Altiplano plateau) in southern Bolivia is an ideal region for
constraining Late-Cenozoic changes in mountain-belt geometry and climate through
94
numerical modeling of foreland-basin stratigraphy because the sediments preserved in the
foreland basin have been well described in the literature and field and laboratory
constraints have been reported in the literature for shortening and exhumation rates in the
hinterland for the period of time over which the eastern margin of the central Andes
formed. A W-E transect of the Cenozoic foreland-basin stratigraphy is exposed within the
retroarc fold-and-thrust belt. Detailed isopach maps and stratigraphic sections for the
Cenozoic foreland-basin stratigraphy have been developed based on a combination of
measured sections from the fold-and-thrust belt and correlations between well log data
and 2-D seismic data (Sempere et al., 1997; DeCelles and Horton, 2003; Echavarria et al.,
2003; Uba et al., 2005; Uba et al., 2006). In addition to data from the foreland basin, the
timing and magnitude of shortening and exhumation within the mountain belt has been
constrained by field mapping and thermochronology (McQuarrie et al., 2002; Muller et
al., 2002; Oncken et al., 2006; Ege et al., 2007; Barnes et al., 2008). At a larger scale, the
Andes as a whole have been an invaluable natural laboratory for exploring feedbacks
between climate and rock uplift (Montgomery et al., 2000; Strecker et al., 2007).
An unresolved issue for the central Andes is: when did the topography of the
central Andes rise to its modern elevation? The classic model for the topographic
development of the central Andes posits that the mean topography reached near-modern
elevation following Neogene crustal thickening within the Subandean zone; thus, earlier
crustal thickening within the Eastern Cordillera was not sufficient for the central Andes
to rise to near-modern elevation (Isacks, 1988; Gubbels et al., 1993). In addition to
crustal thickening, continental lithospheric delamination has been proposed as a
95
mechanism for the rapid Neogene surface uplift posited by the classic model (Kay and
Kay, 1993; Garzione et al., 2008; DeCelles et al., 2009). Continental lithospheric
foundering involves removal of negatively buoyant lower crust (i.e., eclogite root) and
mantle lithosphere. Results from paleoelevation and geomorphic studies indeed support
Late Miocene rapid surface uplift of the region (Ghosh et al., 2006; Hoke et al., 2007;
Garzione et al., 2008). Pedogenic carbonate and carbonate cement samples collected
between 17 and 18°S within the Altiplano and Eastern Cordillera Neogene stratigraphy
contain decreasing δ18O values in progressively younger units (Garzione et al., 2008).
Garzione et al. (2008) interpreted this oxygen-isotope trend as evidence for an increase of
2.5+1 km of the elevation of the Eastern Cordillera during Late Miocene time. Although
these results are in agreement with fossil leaf and clumped isotope data collected from the
Altiplano and Eastern Cordillera (e.g., Gregory-Wodzicki, 1998; Ghosh et al., 2006), the
elevation gain predicted by all three methods can be complicated (i.e., biased toward
larger values) by climate change due to the uplift of the Andes (Ehlers and Poulsen,
2009). Evidence for Late Miocene rock uplift is also recorded by paleosurfaces that exist
on both the eastern and western margins of the central Andes. Barke and Lamb (2006)
estimated 1.7+0.7 km of localized rock uplift for the San Juan Del Oro surface of the
Eastern Cordillera and Interandean tectonomorphic regions since 12-9 Ma when the
surface was abandoned and incised. The Barke and Lamb (2006) results do not constrain
the magnitude of surface uplift, however. The difference between rock and surface uplift
is particularly significant because localized rock uplift can be much higher than mean
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regional surface uplift due to isostatic effects. We will refer to this conceptual model for
surface uplift as the rapid uplift end-member model.
A second end-member model for the topographic evolution of the central Andes
invokes gradual surface uplift since the Late Eocene when deformation propagated from
the Western Cordillera into the Eastern Cordillera. Evidence for pre-Neogene
deformation comes from Eocene exhumation ages within the Eastern Cordillera and
changes in paleocurrent directions within Paleogene stratigraphy of the Altiplano and
Eastern Cordillera (Horton et al., 2002; McQuarrie et al., 2005; Ege et. al., 2007; Barnes
et al., 2008). Although pre-Neogene shortening is considered low (<100-150 km) for
building a high-elevation plateau, long-term shortening should have led to crustal
thickening and isostatic rebound in the Eastern Cordillera. Therefore, unless erosion rates
exceeded or were equal to rock uplift rates, the Eastern Cordillera should have been
rising since the Late Eocene barring some mechanism for keeping the elevation of the
Andes low during an extended period of deformation. The gradual uplift end-member
model posits that the Andes gained the majority of its modern elevation prior to the Late
Miocene (i.e. ~10 Ma).
The Tertiary stratigraphy of the Chaco foreland basin in Bolivia shows an
increase in both grain size and depositional rates during the Late Miocene (Uba et al.,
2006, Uba et al., 2007). This stratigraphic change might be the result of rapid surface
uplift. However, previous studies have inferred that this depositional trend might instead
be primarily controlled by distance from the approaching fold-and-thrust belt (DeCelles
and Horton, 2003; Uba et al., 2006). Thus, the primary mechanism that caused these
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changes in Late Miocene stratigraphy is still uncertain. In addition to thrust belt
migration, the role of climate change and continental lithosphere foundering as controls
on the foreland-basin stratigraphy of the central Andes has also been emphasized in
recent years. For example, Kleinert and Strecker (2001) documented a change from
previously dry to wetter conditions in the Santa Maria basin of northern Argentina
between 9 and 7 Ma. Uba et al. (2007) found a significant (i.e. factor of 5) increase in
time-averaged depositional rates within the stratigraphy of the Subandean thrust belt at
approximately 8-7 Ma based on U-Pb dating of tuffs within Tertiary volcanic rocks.
Coeval with changes in depositional rates and climate conditions within the foreland
depositional facies of the Cenozoic fluvial units shift from single-thread sinuous channels
to more amalgamated alluvial megafan facies (Uba et al., 2006). Uba et al. (2007)
interpreted the change in depositional rates and facies as a consequence of a shift from
semiarid to a more humid climate condition during the onset of the South American
monsoon. An increase in mean annual precipitation should increase sediment supply
through enhanced erosion rates and transport ability of foreland fluvial systems through
higher mean annual discharge.
In addition to thrust belt migration, rapid surface uplift due to crustal thickening
and climate change, lower crustal delamination is another mechanism that has been
proposed for the central Andes to cause surface uplift in the mountain belt and, through
flexural loading of the adjacent foreland basin, modify foreland-basin stratigraphy. The
cordilleran cycle, a conceptual model for mountain-belt development and cyclicity
proposed by DeCelles et al. (2009), posits that as lower crust and mantle lithosphere are
98
underthrust beneath a growing mountain-belt, magmatic and petrologic processes lead to
the formation of a dense eclogitic root that acts as a subsurface negatively-bouyant load,
lowering the elevation of the mountain belt relative to a state of isostatic equilibrium
(DeCelles et al., 2009; Pelletier et al., 2010). If sufficient shortening takes place, the
eclogite root reaches a critical thickness or volume and is delaminated or removed as a
Rayleigh-Taylor instability. This cordilleran cycle, as posited by DeCelles et al. (2009),
includes episodic periods of increased surface uplift and shortening rates, both of which
should have a significant effect on the adjacent foreland basin through the modification
of both rates of sediment supply and creation of accommodation-space. Presently, the
Puna plateau is thought to be in a post-delamination state in this conceptual model
(Schurr et al., 2006; DeCelles et al., 2009) and, as a consequence, has a mean elevation
that is significantly higher than the Altiplano plateau. The Altiplano, in turn, is thought to
have had a delamination event at 10 Ma (Kay et al., 1994). DeCelles et al. (2009) propose
that the Altiplano may already be in an early stage of a new cycle and its lower elevation
is a consequence of newly forming eclogite loads. Evidence for delamination beneath the
Altiplano comes from seismic velocity analysis of the eastern Altiplano lithosphere at
20°S (Zandt and Beck, 2002). A low velocity zone occurs within the upper mantle
beneath the thickened crust of the eastern Altiplano and western edge of the Eastern
Cordillera and is interpreted as a location where the cold-fast upper mantle has been
removed by a delamination event. As such, might stratigraphic trends in the Late
Miocene Chaco foreland-basin deposits be a signature of lithospheric foundering?
99
Previous studies have numerically modeled the evolution of the central Andes as a
coupled mountain-belt-foreland-basin system. Flemings and Jordan (1989) first simulated
the rapid uplift end-member model for the last 5 Myr of deformation in the central Andes
with a two-dimensional model. They concluded that the foreland basin adjacent to the
central Andes should have shifted from narrow and underfilled to broad and overfilled as
sediment supply outpaced sediment accommodation. However, the results of their model
are potentially limited by the fact the mountain belt component of their model was
simplified by assuming a constant topographic slope and sediment supply through time.
Prezzi et al. (2009) recently applied a more rigorous deformation and erosion model to
test the effect of mountain load geometry and elastic thickness of the South American
lithosphere on the sediment accommodation rates within the foreland basin of the central
Andes since the middle Miocene. They found that by decreasing the elastic thicknesses
beneath the Eastern Cordillera and Interandean zones between 14-6 Ma and by deforming
a detailed structural cross section developed by McQuarrie (2002), their model results
adequately fit modern gravity anomalies and isopach distributions for the Late Cenozoic
stratigraphy recently described by Uba et al. (2006) in the Subandean zone.
In this study we explore the linkages between thrust-belt kinematics, climate
change, continental lithosphere delamination, and the foreland-basin stratigraphy in the
central Andes using a coupled, two-dimensional numerical model. Flemings and Jordan
(1989) simulated the foreland response to a fixed topographic slope and constant
sediment supply, and thus, did not capture the effects of feedbacks among mountain-belt
erosion, sediment supply to the foreland basin, and sediment accommodation within the
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foreland basin. The Prezzi et al. (2009) study focused on the changes in foreland basin
accommodation based on surface uplift due to a very specific history of crustal
deformation and changes in elastic thickness under constant climate conditions. Both
studies focused on crustal thickening as the dominant mechanism for driving foreland
basin development. In contrast, in this paper, we aim to constrain the relationship of the
Cenozoic foreland-basin stratigraphy to end-member surface uplift models, climate
change and lower crustal delamination in order to place firmer constraints on the
paleoelevation history of central Andes. First, we determine which end-member surface
uplift model is most consistent with the available stratigraphic and tectonic data for the
development of the eastern margin of the central Andes. Second, we determine whether
there is some signal (e.g., unconformity, grain size change) of climate change or
continental lithospheric delamination recorded in the upper Miocene Chaco forelandbasin stratigraphy.
2. Geologic Background
The central Andes of southern Bolivia contain a high-elevation, internally
draining, low-relief plateau. This portion of the Andes also has the highest magnitude of
total shortening (i.e., approximately 285 km of minimum crustal shortening within the
Eastern Cordillera, Interandean and Subandean zones) (Isacks, 1988; McQuarrie, 2002;
Oncken et al., 2006). The orogenic belt is generally broken up into tectonomorphic
regions which include (from west to east): the Western Cordillera, Altiplano, Eastern
Cordillera, Interandean, Subandean and Chaco Foreland Basin zones (Figure 1). The
modern topographic divide between westward internal drainage into the Altiplano basin
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and eastward drainage into the Chaco foreland basin resides within the Eastern
Cordillera, a bivergent fold-and-thrust belt. Thrusts within the Eastern Cordillera detach
in Ordovician-aged horizons and predominantly exhume Paleozoic through Mesozoic
units (McQuarrie, 2002). Further east, the Interandean and Subandean zones are where
deformation is currently active. These zones contain dominantly eastward-verging
imbricate thrusts that generally exhume younger stratigraphic units compared to the
Eastern Cordillera. The modern deformation front is located between 0.5 and 1 km
a.m.s.l. Beyond the Subandean zone, the Chaco foreland basin extends an additional 250
to 600 km east into Bolivia, Paraguay and the Pantanal wetlands of Brazil until it onlaps
Precambrian basement highs in eastern Bolivia and southwestern Brazil (Horton and
DeCelles, 1997). An exception occurs between 19 and 20°S where the foredeep basin
deposits pinch out onto the Alto de Izozog basement high over a distance that is less than
200 km from the deformation front (Uba et al., 2006).
The Chaco foreland-basin stratigraphy has been documented within the exposed
thrust sheets of the Eastern Cordillera, Interandean and Subandean zones by DeCelles
and Horton (2003), Echavarria et al. (2003), Horton (2005), and Uba et al. (2005).
Isopach trends have also been developed for the buried modern foreland basin units based
on correlations between well data and seismic cross sections (Uba et al., 2006). Starting
at the base of the Cenozoic stratigraphy from the Subandean zone (Figure 2), the Petaca
and Yecua Formations contain fluvial deposits with well developed paleosols,
paleocurrent trends that dominantly show transport toward or along strike with the
approaching mountain belt and thicknesses that are low considering the amount of time
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contained within the units. The Yecua Formation is unique to the Subandean zone
Cenozoic units because it also contains shallow marine facies in addition to fluvial and
lacustrine facies in outcrops located north of 20°S. Moving up through the stratigraphy
into the Tariquía, Guandacay and Emborozú Formations, overall grain size increases
from that of the Yecua Formation, paleocurrent data indicate a shift to transport from the
west, and depositional rates increase. Based on dating volcanic ash layers in the Cenozoic
foreland-basin stratigraphy, there is a factor of five increase in depositional rates between
the Yecua and Tariquía Formations (Uba et al., 2007). Higher in the stratigraphic section,
depositional rates decrease by half upward into the Guandacay Formation. Although
deposition is generally conformable within the Cenozoic stratigraphy, unconformities
bound the Petaca Formation and an angular unconformity separates the Guandacay and
Emborozú Formations. The upper unconformity of the Petaca Formation has been
interpreted to record the uplift and passage of the forebulge as the Subandean zone passed
from the backbulge into the foredeep basin (Uba et al., 2006). In contrast, the angular
unconformity located between the Guandacay and Emborozú Formations has been
interpreted as the initiation of wedge-top deposition in the Subandean zone. Low-angle
unconformities have been documented at the base of coarsening upward cycles within the
age equivalent stratigraphy of the Tariquía and Guandacay Formations in northern
Argentina (Echavarria et al., 2003). However, cycles and significant unconformities were
not documented in the Tariquía and Guandacay Formations of the southern Bolivia
Subandean zone (Uba et al., 2006). Farther west and older in age, similar stratigraphic
trends occur within the remnant foreland-basin stratigraphy of the Eastern Cordillera and
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Interandean zones (Figure 2). The transition from the Cayara into the Camargo Formation
involves an overall increase in grain size, and changes in paleocurrent direction and
apparent depositional rates. One stratigraphic difference between this remnant foreland
basin and the Neogene foreland basin of the Subandean zone is that an unconformity does
not bound the lowest Cenozoic unit in the section. Overall, the most significant trends
within the Cenozoic foreland-basin stratigraphy of the eastern margin of the central
Andes are the apparent increase in depositional rates and grain size with decreasing age,
and thus, any numerical models for the foreland basin evolution of the eastern margin of
the central Andes must honor these trends.
3. Numerical Model Description
The numerical model of this paper is a 2D kinematic model that couples an
actively deforming and eroding mountain belt with sediment transport and flexure within
a depositional foreland basin. A 2D model is valid for this region of the Andes because
the mountain front is nearly linear and paleoflow directions for the majority of the
Cenozoic foreland-basin stratigraphy are perpendicular to the mountain front. The
topography of the numerical model represents the mean topography of the eastern margin
of the central Andes (i.e. western edge of the model begins in the Altiplano) located
between 18-21°S. In our kinematic model, shortening and rock uplift rates are prescribed.
Although time-averaged shortening rates can be directly calculated from published
values, obtaining valid rock uplift rates is an iterative process of running simulations and
checking the results against independent published data (e.g., modern topography and
exhumation magnitudes). In this paper we report model outcomes that are consistent with
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published data for the central Andes, including the modern topography, shortening rates
from balance cross-sections, exhumation magnitudes, Cenozoic isopach data and modern
basin geometries.
3.1. Deformation Model
Dynamic models for crustal deformation (e.g., Simpson, 2004) that directly solve
for visco-plastic deformation within a fold-and-thrust belt could be applied to the central
Andes. This approach, however, comes with the drawbacks of longer computational
times and the difficulty of calibrating the model to multiple types of calibration data. We
therefore take a simpler approach in this paper. In the model, the mountain belt is
partitioned into individual blocks for each tectonomorphic region (i.e., Altiplano, Eastern
Cordillera, etc.). Exceptions are made for the Eastern Cordillera and Subandean zones.
The Eastern Cordillera, a bivergent fold-and-thrust belt, was divided into separate
backthrust and forethrust blocks because these two regions rapidly exhume at different
periods of time (Barnes et al., 2008). The Subandean zone was divided into a western and
eastern block to allow the central and eastern Subandean zone to continue to subside until
deformation propagated into the region during the Late Miocene. Deformation is
simulated by uniform rock uplift and shortening applied to each actively deforming
block. Time-averaged shortening rates were determined for input to the model by
dividing the total shortening determined from field-based balanced cross sections for
each tectonomorphic zone by the duration of fault activity (McQuarrie, 2005; Muller,
2002). The positions of the model nodes located to the east of the deformation front are
fixed while the model nodes to the west of the deformation front are allowed to translate
105
toward the foreland basin as if riding over a décollement. The total propagation of the
deformation front (and hence the forebulge) is approximately 536 km, which is close to
the approximately 600 km of total propagation of the forebulge since the Late Eocene
estimated by DeCelles and Horton (2003).
3.2. Erosion Model
A stream-power model is used to determine bedrock incision rates within the
mountain belt (Howard and Kirby, 1983; Whipple and Tucker, 1999):
∂h
∂h
= −k e A m
∂t
∂x
n
= −k e A1 / 2 S = −k e lS
(1)
where h is elevation a.m.s.l. (m), t is time (yr), ke is the bedrock erodibility (yr-1 for n =
1.0 and m = 0.5), A is the contributing drainage basin area (m2), x is the lateral distance
from the model origin (m), l is the distance along the principal channel from the
headwaters of the drainage basin (m), S is the local channel slope (unitless) and m and n
are constants that determine the dependence of local erosion on discharge and channel
slope. Equation (1) determines the erosion rates of the major streams that act as the local
base level for the mountain front topography. In equation (1) we assume that the length of
the principal channel is proportional to the square root of the contributing drainage area
(Hack, 1957). We also assume that the area and slope exponents m and n have values of
0.5 and 1 respectively. Evidence from theory and field studies predicts that the ratio of
m/n is near 0.5 (Whipple and Tucker, 1999). Although the slope exponent n can range
between 0.66 and 2.0 depending on the relationship between slope and stream power
106
(shear stress), we chose to make the stream power linearly proportional to the slope (n =
1.0 and m = 0.5), consistent with the assumption of many other studies (e.g., Kirby and
Whipple, 2001; Snyder et al., 2000). Bedrock incision rates for active mountain belts are
on the order of 0.1 to 1.0 mm/yr (Montgomery and Brandon, 2002). When m/n is equal to
1/2, we found that ke must be on the order of 10-6 (m/yr) to appropriately reproduce the
range of calculated exhumation rates (i.e., 0.1-0.6 mm/yr) for timescales of 102 to 107
years for the central Andes (Safran et al., 2005; Ege et. al., 2007; Barnes et. al. 2008).
The boundary between the bedrock portion of the model (where stream-power-driven
erosion of bedrock is modeled) and the alluvial portion of the model (where grains-sizedependent sediment transport is modeled) is allowed to fluctuate in the model. If a given
model node contains sediment or sediment is deposited (i.e., upstream sediment supply
exceeds the transport capacity), then the node is treated as an alluvial channel and no
bedrock incision occurs. Otherwise, bedrock incision occurs and the amount of newly
created sediment transported downstream is controlled by the transport capacity.
3.3. Sediment Transport Model
Sediment transport of a bimodal grain size distribution (i.e., gravel and sand) is
simulated with a modified version of the diffusion model approach, which states that
sediment flux is proportional to local channel slope. Linear slope-dependent sediment
flux, when combined with conservation of mass, gives a diffusion equation for the
evolution of the longitudinal profile of the foreland basin (Paola et al., 1992). In the
model of this paper we added a threshold slope term to the diffusion equation model
because transport of gravel requires a threshold slope to initiate transport. The equation
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for alluvial deposition and erosion along a channel profile in our model is a combination
of mass balance (i.e., the Exner equation) and the slope-dependent sediment-transport
equation:
qs = k g (S − φ ) , S > φ
qs = 0
,S ≤φ
qs, upstream − qs, downstream
∂q
∂h
=− s =
∂t
∂x
∂x
(2)
(3)
where kg is the transport coefficient (m2/yr), φ is the threshold slope which is a function
of grain size and qs is the local sediment flux (m2/yr). The transport coefficients for
gravel and sand are determined from a relationship that depends on discharge and river
type (Paola et al., 1992). Paola et al. (1992) calculated the values for braided and
meandering rivers to range between 1.0-7.0 x 104 (m2/yr) for a drainage basin with a
length of 100 kilometers and a mean annual precipitation of 1 m. We chose a value of 1.0
x 104 (m2/yr) for the gravel transport coefficient, which is a reasonable value for braided,
gravel-dominated streams with relatively small drainage areas. The transport coefficient
for sand was chosen to be approximately 6.5 x 104 (m2/yr), which is on the order of the
transport coefficient used by Flemings and Jordan (1989) for their simulation of the
central Andes. The threshold slope for gravel entrainment is approximately 0.001, which
represents effective channel parameters of a median grain size of 0.02 m and bankfull
flow depth of 1.85 m. The chosen median grain size is on order with the median grain
size of bedload material observed within the modern Rio Pilcomayo located in the
Eastern Subandean zone (Mugnier et al., 2006).
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Channel armoring is an emergent property of the model obtained by incorporating
a threshold slope for gravel transport. If discharge is insufficient to entrain gravel present
on the channel bed during a time step, then the finer-grained sand deposited beneath the
gravel is prevented from being transported during that time step. This process should
reduce both alluvial and bedrock erosion in regions of lower regional slope where gravel
is present. Active bed material layers in natural channels are on the order of 1-2 grains
thick (Hassan et al., 2006). Although our active layer is on the order of a meter, it is valid
to have a larger active layer for the purposes of this study because we are focusing on
long-term (>105 yr) grain size trends that average over many individual flooding events.
3.4. Basin Flexure
The bedrock and alluvial surface dynamics models are coupled to a flexural
foreland basin in order to quantitatively assess accommodation-space creation and the
migration of the forebulge through time in the model. The flexural model solves for the
displacement of a thin elastic beam subjected to a spatially-distributed vertical load
(Turcotte and Schubert, 2002):
D
∂ 4 w( x )
+ (ρ m − ρ s ) gw( x ) = L( x)
∂x 4
(4)
where w is the deflection of the Earth’s crust (m), D is the flexural rigidity (Nm), ρm is
the density of the mantle (kg/m3), ρs is the density of the mountain crust or foreland
sediment (kg/m3), g is the acceleration due to gravity (m/s2), and L(x) is the topographic
load (kg/ms2). For each simulation, except for those involving eclogite root delamination
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(discussed later), we solve for the deflection due to a topographic load every time step
using a Fourier transform method (Watts, 2001). Viscous relaxation effects were not
considered in the model because we interpret stratigraphic patterns over geologic
timescales that are greater than relaxation timescales.
Several studies have calculated the flexural rigidity of the central Andes using 2dimensional methods (Horton and DeCelles, 1997; Stewart and Watts, 1997; Tassara,
2005). Their results suggested that flexural rigidities along the eastern margin of the
central Andes range between 1.5x1023 and 4.0x1024 Nm. We chose to use a flexural
parameter of 150 km because this value best fits the observed modern basin geometry
between 18-20°S. This value is consistent with the results of Chase et al. (2009), who
found that the flexural parameter should be less than 220 km for the central Andes. The
flexural parameter is defined as the following (Turcotte and Schubert, 2002):
⎡
⎤
4D
α =⎢
⎥
⎣ (ρ m − ρ s )g ⎦
1/ 4
(5)
where α is the flexural parameter (km). Rearranging equation (5) and applying values
from Table 1 yields a flexural rigidity of approximately 6.8x1023 Nm, which is well
within the range of calculated effective flexural rigidities for the central Andes. Although
previous studies have suggested that the elastic thickness varies in space and time (e.g.,
Toth et al., 1996; Prezzi et al., 2009), such variations are difficult to constrain. Therefore,
our model implements a uniform and constant elastic thickness.
3.5. Numerical Modeling Methods
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We ran three types of experiments using our numerical model: (1) an end-member
surface-uplift model experiment, (2) an eclogite-root-foundering experiment, and (3) a
climate-change experiment. The end-member surface-uplift model experiment simulates
the last 43 Myr when deformation is concentrated in the eastern margin of the central
Andes. The purpose of this experiment is to contrast the foreland basin response (e.g.,
changes in depositional rates and grain size) to a mountain belt that is gradually uplifting
through time against the foreland-basin response to rapid surface uplift caused by
Neogene crustal thickening. The parameters used in this experiment are found in Table 1.
During the simulations of this experiment, climatic conditions were held constant and
time steps are approximately 700 yr. Topographic profiles were sampled at 22, 10 and 0
Ma for visual comparison.
Following the end-member surface-uplift model experiment, we ran an eclogiteroot-foundering experiment. The purpose of this experiment is to contrast the forelandbasin response to rapid surface uplift in the mountain belt caused by crustal thickening
alone against rapid surface uplift caused by a combination of foundering and crustal
thickening. The model duration, time steps and interval of topographic profile sampling
in the foundering experiment are the same as in the end-member surface-uplift
experiment. We also apply the parameters in Table 1 and the same kinematic histories in
the mountain belt (e.g., shortening rates and propagation of deformation) as in the rapid
uplift model. An eclogite root grows in our model between 25 and 10 Ma beneath the
Altiplano and Eastern Cordillera backthrust zones where significant crustal thickening
has taken place such that lower crustal rocks are subjected to pressures sufficient enough
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to produce eclogite. We assume that eclogite foundering is caused by a Rayleigh-Taylor
instability (defined as the diapiric drip of a dense layer overlying a less dense layer) and
that the timescales over which the growth of the eclogite drip occurs can be calculated to
within first order by the results of the linear-stability analysis of Turcotte and Schubert
(2002):
τa =
13.04 μ
(ρ e − ρ m )b
(6)
where τa is the amount of time required for an instability to grow by a factor of e, μ is the
viscosity of the upper and lower layers and b is the original thickness of the eclogite
layer. The Rayleigh-Taylor instability is not modeled explicitly but instead we prescribe a
rate of eclogite root growth consistent with the observed spacing between foundering
events in the central Andes and the thickness of the root required to initiate foundering.
Equation (6) is rearranged to solve for b in order to prescribe the thickness of the eclogite
root required to initiate foundering in our model. Based on geophysical models and
paleoelevation proxies, the length of time over which the foundering event occurred
beneath the Altiplano was approximately 3 Myr (i.e. between 10 and 7 Ma) (Molnar and
Garzione, 2007; Garzione et al., 2008). We assume that the majority of the foundering
time is spent growing the instability in the lower crust-mantle interface by the initial
factor of e due to the high resistance to flow of the mantle when minimal perturbations in
the lower crust-mantle interface exist. Later, when a significant pressure gradient exists at
the base of the eclogite layer due to the formation of a drip; the pinching and foundering
portion of the drip event occurs much more rapidly. Based on this argument we inferred
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that τa is approximately less than or equal to 3 Myr. Using densities of 3300 and 3600
kg/m3 for upper mantle and eclogite and a root growth period of 3 Myr, the range in
maximum eclogite layer thickness is calculated to be between 1.87 to 46.88 km as the
effective viscosity of the underlying mantle layer varies between 4.0 x 1019 and 1.0 x 1021
Pa s. This range in eclogite layer thickness is consistent with the 10 to 20 km eclogite
root thickness that occurred prior to removal during a numerical simulation of mantle
drip for the Puna plateau (Quinteros et al., 2008). We used a similar scale (i.e. 12.5 km)
for the thickness of the eclogite root required to initiate delamination.
Between 10 and 7 Ma, the root is removed at a linear rate until it is completely
removed at 7 Ma. During the period of root foundering, we allowed our flexure algorithm
to specify the magnitude of rock uplift due to foundering and superimposed that result on
the results of the rapid-uplift model. Our model is kinematic, so the viscous coupling
between the sinking root and the overlying lithosphere is not simulated in our model.
However, there is a possibility that viscous coupling may result in significant (i.e., on the
order of 100s meters) subsidence and rebound (Gogus and Pysklywec, 2008).
An experiment with climate change during the late Miocene was also conducted
in order to determine its impact on the foreland-basin stratigraphy. At 9 Ma, we simulated
the onset of the South American monsoon by increasing both the bedrock erodibility and
sediment transport coefficients in response to increased mean annual discharge and storm
intensity. The bedrock erodibility (ke) of the stream power model and the transport
coefficients (ks and kg) of the diffusion model are both theoretically shown to be
proportional to discharge (Paola et al., 1992; Whipple and Tucker, 1999). If we assume
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that discharge scales proportionally with precipitation, then erosion rates and sediment
transport rates scale proportionally with precipitation. This is reasonable assumption for
erosion rates because long-term erosion rates have been shown to correlate with mean
annual precipitation rates in the Andes (Kober et al., 2007). Sediment transport rates in
alluvial rivers have been proposed to correlate with mean annual precipitation (Molnar et
al., 2006). Although the magnitude of increase in mean annual precipitation at the onset
of the South American monsoon unknown, long term erosion rates since the Eocene have
not varied by more than a factor of two from modern rates (Safran et al., 2005; Ege et. al.,
2007; Barnes et. al. 2008). Thus, both erosion rates and sediment transport rates increase
by a factor of two at 10 Ma. The model duration, time steps and interval of topographic
profile sampling are the same as in the end-member uplift models.
4. Numerical Modeling Results
4.1. End-member surface-uplift model experiment summary
Topographic-profile and sediment-flux time-series plots show the development of
the eastern margin of the central Andes in response to the gradual uplift end-member
model (Figures 3A and 3B). Between 43 and 22 Ma, deformation is concentrated within
the Eastern Cordillera. During this period of time, topographic slopes are low, and thus
the sediment fluxes from the mountain belt into the Altiplano and foreland basin are also
low. Sediment eroded off of the Eastern Cordillera is not transported beyond the
foredeep, which is underfilled to completely filled. The sediment flux leaving the
backbulge basin toward the east is zero at this time. The Subandean zone in its predeformed state is located more than 400 km from the deformation front near the
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forebulge crest (Figure 3A). Low sediment flux from the mountain belt is consistent with
observed paleocurrent data from the Petaca Formation, located in the Subandean zone,
showing westward paleotransport from the South American craton. Between 30 and 22
Ma there is a period of higher than average sediment flux (Figure 3b). Sediments with
low erodibility contained within wedgetop basins are exhumed as the eastern portion of
the Eastern Cordillera begins to uplift at 30 Ma. Exhumation of the wedgetop basin
deposits causes the period of high sediment delivery to the foredeep.
By 22 Ma, rock uplift rates between 0.1 and 0.2 mm/yr cause the Eastern
Cordillera to rise to a peak elevation of 3 km a.m.s.l. The increase in topographic relief
causes the mean sediment flux from the mountain belt into the foreland basin to increase
such that sediment from the mountain belt outpaces foreland accommodation rates and is
transported out of the foredeep. Subsidence rates in the backbulge basin are an order of
magnitude lower than in the foredeep. Consequently, the backbulge basin is rapidly filled
and sediment from the mountain belt begins to bypass the filled backbulge basin by 27
Ma (Figure 3B). At this time the deformation front propagates east into the Interandean
zone and creates a step in topography located 500 km from the left end of the model
domain in Figure 3A. Again, there is a period of higher-than-average sediment flux
between 22 and 15 Ma caused by exhumation of wedgetop basins within the Interandean
zone. The 22 Ma snapshot represents the time period when the Camargo Formation is
transitioning into the overlying Mondragon Formation of the Eastern Cordillera and when
the Petaca Formation is being deposited in the Subandean zone.
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By 10 Ma, the core of the Eastern Cordillera is within 1 km of its modern peak
elevation and the deformation front and forebulge migrate another 100 km toward the
South American craton (Figure 3A). As a result of the increase in relief and mountain
front slopes, the average sediment flux into the foredeep increased from 20 to 30 m2/yr
and overfills the foredeep. A plot of sediment flux leaving the backbulge basin shows that
at least half of the mean sediment supply to the foreland is leaving the backbulge basin at
this time (Figure 3B). A significant decrease in the amount of sediment leaving the
backbulge basin and a significant increase in the amount of sediment entering the
foredeep occurs between 5 and 2 Ma. This is the period in time when the eastern
Subandean zone is actively uplifting. High rock uplift rates in the eastern Subandean zone
lead to exhumation of wedgetop basins and the development of a topographic step.
Although sediment flux from the mountain belt is high during this time, rapid creation of
topographic relief in the front of the thrust belt causes high subsidence rates in the
foredeep. A significant portion of the sediment supply to the foreland basin is deposited
and stored in the foredeep. The final topography at 0 Ma fits the average and maximum
topography of the modern central Andes well. However, the alluvial slopes of the
depositional foreland basin are a few hundred meters higher than the average channel
elevations today. It is unclear if this error is due to an over prediction of the sediment
supply, an under prediction of transport coefficient, or if sediment fluxes at the
downstream end of the model domain are too low.
Although rates of shortening and the propagation of the deformation front are
identical for both the gradual and rapid uplift models, the surface uplift and sediment flux
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response to the rapid uplift model is notably different (Figure 4A and 4B). Between 43
and 22 Ma, low rock uplift rates (i.e., 0.1-0.2 mm/yr) in the Eastern Cordillera lead to
only 1.7 km of peak surface uplift over this period of time. As such, mean sediment
supply rates from the mountain belt remain low (i.e., <10 m2/yr) over the first 21 Myr of
simulation. Despite the fact that the mean sediment flux into the foreland is half the rate
of the gradual uplift model, the foredeep and backbulge basins in the rapid uplift model
are able to fill and bypass sediment from the mountain belt by 25 Ma. Following an
additional 12 Myr of active uplift between 22 and 10 Ma, peak elevation of the Eastern
Cordillera in the rapid uplift model rises to 2.3 km a.m.s.l. and deformation propagates
east into the western Subandean zone. Mean sediment flux into the foreland basin and
sediment fluxes leaving the backbulge basin in the rapid uplift model remain low
compared to the values for the gradual uplift model at this time. At 10 Ma, rock uplift
rates across the mountain belt increase from 0.1 to 0.3-0.5 mm/yr as the central Andes
rapidly uplift in response to deformation in the Subandean zone. As a result, mean
sediment flux from the mountain belt increases from 20 to 60 m2/yr. However, the
sediment flux leaving the backbulge basin decreases shortly after the initiation of rapid
uplift across the mountain belt because subsidence rates in the foredeep quickly respond
to surface uplift within the mountain belt, trapping sediment within the foredeep near the
edge of the mountain belt. The final topography in the rapid uplift model at 0 Ma equally
reproduces the modern mean and maximum topographies of the central Andes mountain
belt and foreland basin. Again, the predicted foreland topography is slightly higher than
that of the observed.
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4.2. Constrains on foreland basin depositional rates
In both the gradual and rapid uplift models, surface uplift of the eastern margin of
the central Andes leads to an increase in sediment supply and subsidence within the
foreland basin. Figures 5A and 5B show the sediment thickness curves for four
depozones (i.e., the forethrust region of the Eastern Cordillera, western Interandean zone,
western Subandean zone and eastern Subandean zone, respectively) within the foreland
basin for both the gradual and rapid uplift models. The solid lines represent the results for
the simulations described in the previous section. Between 43 and 22 Ma deformation
and surface uplift is limited to the Eastern Cordillera. At this time, both the eastern region
of the Eastern Cordillera and western Interandean zones are located within the foredeep
basin of the gradual uplift model (Figure 5A). Between 40 and 30 Ma the deformation
front remains in the core of the Eastern Cordillera. A spatially-fixed load leads to
constant subsidence rates, and thus, constant depositional rates in the Eastern Cordillera
depozones over this period of time. The Interandean depozone, however, is located in the
distal foredeep, and therefore develops an acceleration in depositional rates between 31
and 25 Ma that is characteristic of an increase in subsidence rates due to an approaching
mountain belt. Further to the east, the western and eastern Subandean depozones are
located within the backbulge basin. Low subsidence rates lead to low depositional rates
between 43 and 36 Ma. Between 36 and 25 Ma, the forebulge crest migrates through the
western Subandean depozone and leads to erosion and development of an unconformity.
Sediment deposited while the eastern Subandean depozone was located in the backbulge
is completely eroded away during this time period. The shaded region between 36 and 25
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Ma represents the approximate depositional age for the Camargo Formation. Predicted
depositional thicknesses for the Camargo Formation range between 1 and 2 km in the
Eastern Cordillera and Interandean depozones, but under predict the > 2 km thickness of
the Camargo Formation observed in the Camargo syncline (DeCelles and Horton, 2003).
By 22 Ma, deformation propagates into the Interandean zone. Therefore, the Interandean
depozone is uplifted and exhumed at this time. Between 22 and 10 Ma, depositional rates
within the western Subandean zone accelerate as it reaches the foredeep. At 9 Ma, a sharp
inflection point occurs that reflects a significant increase in depositional rates in the
eastern Subandean zone. These high depositional rates are inferred to be the result of high
rock uplift rates in the western Subandean zone. Gradual uplift model simulations were
also conducted for constant and lower rock uplift rates for the western Subandean zone.
However, a slow and gradual kinematic model for the deformation front could not
reproduce the extreme increase in isopach thickness for the Tariquía Formation given the
constraints on deformation propagation rates during the Late Miocene. Reproducing
accurate accommodation magnitudes for the foreland-basin stratigraphy while also fitting
the modern topography motivated applying an increase in rock uplift rates at the front of
the mountain belt to the gradual uplift model. Rock uplift rates within the backthrust and
forethrust of the Eastern Cordillera, however, remained gradual and constant. Increased
uplift rates between 9 and 4 Ma caused an increase in sediment accommodation rates. As
a result, a larger portion of the incoming sediment flux is stored in the foredeep
depozone. Extremely high depositional rates continue until approximately 4 Ma when the
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basin begins to uplift and exhume. This period of time represents the deposition of the
Guandacay Formation and the transition into the Emborozú Formation.
Although shortening rates and thrust belt propagation rates in the rapid uplift
model are the same as in the gradual uplift model, the total thicknesses of basins for the
rapid uplift model are significantly different than those for the gradual uplift model
(Figure 5B). Early in the model experiment between 43 and 22 Ma, lower rock uplift
rates within the Eastern Cordillera cause depositional thicknesses within the eastern
forethrust region of the Eastern Cordillera and Interandean zones to be 500 m less
compared to thicknesses predicted in the gradual uplift model. Both locations develop
Camargo Formation thicknesses of approximately 500 m, which is much less than the
observed > 1 km thickness for the Camargo Formation in the Camargo syncline. By 12
Ma, low rock uplift rates in the Eastern Cordillera and Interandean zones lead to a factorof-2 decrease in the total sediment thickness of the western Subandean depozone
compared to the thickness predicted for this depozone in the gradual uplift model. At 9
Ma, the entire mountain belt between the western Subandean zone and Altiplano rapidly
uplifts. This rapid uplift causes an order of magnitude increase in depositional rates
within the eastern Subandean depozone. Approximately 2 km of sediment are deposited
between 8-6 Ma. However, this is still 1 to 1.8 km less than the maximum isopach values
reported by Uba et al. (2006) for the Tariquía Formation.
Additional simulations were conducted for both surface-uplift models to test the
sensitivity of the sediment thickness curves to the rigidity of the South American plate.
The dotted lines represent the results for simulations with the lowest rigidity calculated
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for the South American plate (i.e., 1.5x1023 Nm). Decreasing the rigidity in the gradual
uplift model simulation does not significantly affect the depositional thicknesses within
the Eastern Cordillera (Figure 5A). However, the maximum thicknesses of the other three
depozones decrease by approximately 300-500 m. Conversely, increasing the rigidity of
the South American plate (dashed lines) to the upper limit of rigidities previously
calculated for the South American plate (i.e., 4.0x1024 Nm) causes the maximum
depositional thickness of three western most depozones to increase with respect to the
initial results. Changing the rigidities of the South American plate in the rapid uplift
model has similar effects as in the gradual uplift model (Figure 5B). Although the rigidity
of the South American plate can range over an order of magnitude, the predicted
thickness of the Camargo Formation within Eastern Cordillera and Interandean
depozones of the rapid uplift model is less than a kilometer.
4.3. The role of eclogite-root-foundering on surface uplift and foreland development
The eclogite-root-foundering simulation involves the growth and removal of a
subsurface load that modifies the flexural response of the foreland basin and rock uplift
distribution within the mountain belt. Prior to 10 Ma, rock-uplift rates and shortening
rates are prescribed to be identical to the rapid-uplift model for the region of the
mountain belt that is deforming. Between 25 and 10 Ma, an eclogite load is allowed to
uniformly grow beneath the Altiplano and backthrust region of the Eastern Cordillera
where the deformed crust is sufficiently thick for lower crustal rocks to undergo a phase
transition to eclogite (Figure 6A). The subsurface eclogite load, whose location is
represented by the black bar, sets up a flexural profile (identified by the thick black line)
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that superimposes onto the flexural profile caused by the topographic load. Although the
total deflection due to the presence of the eclogitic root is small (i.e. ~1m) at 25 Ma, by
10 Ma the total deflection is on the order of 1 km at the center the eclogite root. At 25
Ma, the edge of the eclogite load is located approximately 150 km from the current
deformation front. As a result of the distance between the load and deformation front,
subsidence caused by the eclogite load is small in the foredeep and shortly to the east
subsidence changes to rock uplift in the distal foredeep. As time progresses, the
Interandean and Subandean depozones are subject to an additional component of rock
uplift because they are located on the flexural forebulge related to the eclogite root load.
Following 10 Ma, the polarity of lithospheric deflection reverses as the eclogite root is
removed (Figure 6B). The solid black curve is the total amount of deflection that occurs
over 3 Myr as the subsurface load is removed. A significant amount of rock uplift occurs
directly over the center of the load in the eastern Altiplano and western backthrust region
of the Eastern Cordillera and rapidly decays into the core of the Eastern Cordillera.
Further east, the Interandean and western Subandean zones experience subsidence on the
order of meters to tens of meters. Beyond the western Subandean zone, the deflection due
the eclogite foundering is less than a meter. Rock uplift rates in the Altiplano and the core
of the Eastern Cordillera due to the eclogite root foundering range between 3.0-6.0x10-4
m/yr. Maximum deflection rates in the eastern Interandean and western Subandean zones
are on the order of 10-7 and 10-5 m/yr. Rock uplift rates near the mountain front and
within the foreland basin (i.e., Interandean and Subandean zones) prescribed in the rapid
uplift model at 10 Ma are on the order of 6.0-5.0x10-4 m/yr, and therefore, the flexural
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signal from a delamination is small when compared to the subsidence or rock uplift due
to crustal thickening. Conversely, rock uplift rates due to the delamination are slightly
greater in the Altiplano and the backthrust region of the Eastern Cordillera than the
prescribed uplift rates due to crustal thickening.
Sediment supply into the foredeep predicted by the eclogite-root-foundering
model shows a similar behavior to that of the rapid-uplift model. Sediment supply rates
are low until approximately 10 Ma when the eclogite drip event accompanied by the
rapid uplift of the eastern Interandean and western Subandean zones lead to increased
channel slopes in the front of the mountain belt (Figure 7A). The amount of sediment
bypassing the foreland basin in the eclogite-root-foundering model is also similar to the
rates predicted by the rapid uplift model with the exception of a greater magnitude of
sediment flux between 15 and 5 Ma. This increase in sediment leaving the foreland per
time is either the result of increased erosion rates in the mountain belt or decreased
sediment accommodation rates in the foreland basin. A comparison of sediment
thicknesses predicted by the rapid uplift and eclogite-root-foundering models through
time for the Interandean, western Subandean and eastern Subandean zones shows that the
drip event modified sediment accommodation by less than 10 percent of the maximum
basin thicknesses before foundering (Figure 7B). The western Interandean zone basin
deposits are approximately 100 meters thicker at 22 Ma for the eclogite-root-foundering
model (solid line) compared to the rapid uplift model (dashed line). Between 25 and 22
Ma the western Interandean zone is located close enough to the edge of the eclogite load
to experience subsidence which causes the change in sediment thickness between models.
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Conversely, the Subandean basins are located far enough from the eclogite load at that
time to undergo rock uplift. The western Subandean basin is the least affected basin by
the drip event because of its proximity to the inflection point between subsidence and
rock uplift caused by accumulation of eclogite. The eastern Subandean zone also
experiences rock uplift due to foundering, and thus, lesser sediment thicknesses between
20 and 10 Ma. Decreased accommodation in the Subandean zones leads to an increase in
sediment leaving the basin prior to 10 Ma. Following 10 Ma, the drip event leads to
minor subsidence in the eastern Subandean zone and the sediment thickness differences
between the two models becomes negligible.
4.4. The role of climate change in foreland basin development
The results for the climate change experiment that involves a factor-of-two
increase in mean annual precipitation during the Late Miocene due to the onset of the
South American monsoon are reported here. Bedrock uplift rates during the Late
Miocene within the actively deforming mountain belt in this simulation are increased
with respect to those of the rapid-uplift model to maintain similar surface uplift histories,
and thus, the basin accommodation histories between the rapid uplift model and the
climate change model are the same. We only report observations for sediment flux and
grain size. Although erosion rates and sediment transport rates are a factor of two lower
for the first 35 Myr, the first order behavior of sediment flux entering the basin appears to
be unchanged from that of the rapid uplift model (Figure 8). The effect of lowering both
erosion and sediment transport coefficients by a factor of two early on in the simulation is
a few million years delay in the time when sediments begin to exit the backbulge basin.
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Decreasing both erosion rates and sediment transport rates also has a cumulative effect of
decreasing sediment bypass rates (solid black line) by almost half the value of sediment
bypass predicted by the rapid uplift model (dashed line). Following 9 Ma, the sediment
bypass rates suddenly increase by a factor of 1.5 times the previous flux.
An abrupt change in erosion rates and transport efficiency due to climate change
should be expected to have an effect on grain size within the depositional basin, and
therefore, we tracked the gravel-sand interface for each of the experiments (Figure 9). A
comparison of the gradual uplift, rapid uplift and climate change model results (Figure 9)
reveals that in general the gravel-sand boundary closely tracks the location of the
deformation front (shown as the dashed line). An exception to this observation occurs
immediately following times when the deformation front propagates basinward and the
gravel-sand interface lags behind the deformation front for approximately 1-2 Myr until
the regional slope of the newly formed wedgetop basin increases above the threshold
slope for gravel transport. Once gravel reaches the deformation front, gravel progradation
into the foredeep appears to be limited within a narrow zone of approximately 50 km
from the deformation front. Figure 9A compares the gravel-sand boundaries for the
gradual and rapid uplift models. Prior to 10 Ma, the mean location of the gravel-sand
boundaries for both models generally remain in front of the deformation front following
the 2 Myr lag periods. However, the initiation of rapid uplift in the Interandean and
western Subandean zones at approximately 10 Ma causes the gravel-sand boundary to
retrograde back into the wedgetop zone as accommodation rates exceed sediment supply
rates. Both models appear to closely overlap each other with two exceptions. Between 22
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and 15 Ma, the gravel front in the gradual uplift model progrades more rapidly than the
gravel front of the rapid uplift model due to a combination of greater uplift rates in the
Interandean zone wedgetop basin and greater sediment supply rates. Between 8 and 5
Ma, the mean location of the gravel-sand interface of the rapid uplift model retrogrades
more than the gravel-sand interface of the gradual uplift model due to greater subsidence
rates. Figure 9B compares the results for the climate-change model to the rapid-uplift
model, and thus, should highlight the effects of changes in precipitation on grain size.
Although both the bedrock erodibilities and transport coefficients are a factor of two less
in the climate change model compared to the rapid-uplift model between 25 and 9 Ma,
the gravel-sand boundaries generally track each other well through time. However, there
are a few brief periods of time when the gravel-sand interface of the climate change
model lags behind that of the rapid uplift mode. When the bedrock erodibilities and
transport coefficients increase by a factor of root 2, the results of the climate change
model do not vary significantly from the trends of the rapid-uplift model.
5. Discussion
Two end-member surface-uplift models have been proposed for the central Andes
(Garzione et al., 2008). One model involves a long history of constant deformation since
the Late Eocene, accompanied by a gradual increase in mean surface elevation prior to 10
Ma. The opposing end-member model suggests that the central Andes did not gain
significant elevation until after 10 Ma. Our results show that if the elastic properties of
the underthrusted South American plate remain effectively constant, then rapid rock
uplift is likely to have occurred within the Interandean and western Subandean zones
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during the late Miocene in order to generate sufficient accommodation space for the
Tariquía Formation observed in outcrops within the central to eastern Subandean zone
(Figure 5). Constraints on the initiation of rapid rock uplift in the Interandean and
Subandean zones is based on the isopach data for the Yecua and Petaca Formations. If
rock uplift rates were greater in the Interandean and Subandean zones prior to 9-8 Ma
when the Tariquía Formation began to be deposited, then the isopach trends for the
Yecua and Petaca Formations observed within the central Subandean zone would show
maximum thicknesses greater than 825 meters. Rock uplift rates prescribed for the
Interandean and western Subandean zones in the gradual uplift model lead to maximum
thicknesses for the Petaca-Yecua deposits that are on the order of 1 km. Although other
studies have proposed that decreasing the flexural rigidity of the South American
lithosphere can explain an increase in accommodation for the Tariquía Formation (Prezzi
et al., 2009), we show that a constant flexural rigidity model can also fit the Late
Miocene isopachs well.
At 10 Ma, both the Altiplano and Eastern Cordillera are sufficiently far from the
central and eastern Subandean zones such that their contribution to the accommodation
space is low compared to that of the Interandean and western Subandean zones. The
deflection of an elastic beam in response to a point load decreases exponentially with
distance from the center of the load. As such, a comparison of the sediment thickness
time series results for the gradual and rapid uplift models shows little change in
accommodation rates following 10 Ma in the eastern Subandean zone even though the
peak elevations of the Eastern Cordillera differ by 2 kilometers (Figure 5). Although the
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difference in peak elevation within the Eastern Cordillera does not strongly affect Late
Miocene foreland-basin sediment accommodation, we can constrain the early surface
uplift history of the Eastern Cordillera and Altiplano with the middle Tertiary
stratigraphy of the Eastern Cordillera and western Interandean zones because these
depozones were located at least 100 km closer to the Eastern Cordillera than the central
Subandean zone. DeCelles and Horton (2003) measured a thickness of over 2 km for the
Camargo Formation within the Camargo Syncline of the Eastern Cordillera. A
comparison of sediment thicknesses deposited between 36 and 22 Ma predicted by the
gradual and rapid uplift models shows that a sediment accommodation thickness of 2 km
within the early foreland basin is not achievable unless rock uplift rates in the core of the
Eastern Cordillera more closely resembled the rock uplift rates specified within the
gradual uplift model (Figure 5). Although there is uncertainty in the depositional age of
the Camargo Formation, the rapid uplift model could not create sufficient sediment
accommodation space if deposition began as early as 40 Ma. If the erosion rates in our
model are close to the effective erosion rates during the Early Miocene and Late
Oligocene, then the thick deposits of the Camargo Formation would imply that the peak
elevation of the Eastern Cordillera was between 2-3 km by 22 Ma.
An Eastern Cordillera peak elevation of 2-3 km might have acted as an effective
topographic barrier to moisture that was being transported west across the central Andes.
Today, the topography of the eastern flank of the central Andean Plateau prevents a
significant amount of moisture that originated from the Atlantic Ocean and Amazon
Basin from being transported west into the Altiplano and Western Cordillera by the South
128
American Summer Monsoon (Strecker et al., 2007). A similar behavior occurs in the
southern Andes where the moisture from the Pacific Ocean carried by the Southern
Hemisphere Westerlies rains out on the west coast of South America and the Patagonian
Andes leading to semiarid conditions on the eastern flank of the Andes. The southern
Andes are an effective moisture barrier even though their mean elevation is around 2 km
lower than the central Andes with peak elevations ranging from 2-3 km between 38 and
50°S. We envision the topography of the central Andes during the early to middle
Miocene as being similar to the southern Andes today, which suggests that the orogen
may have been an effective barrier to moisture being transported to the southwest from
the Atlantic. If this is the case, then basins within the Altiplano and Western Cordillera
regions of the central Andes should become increasingly arid around or soon after 22 Ma.
Based on abrupt changes in lacustrine facies within basins, the onset of hyper-aridity has
been documented to occur between 10 and 6 Ma for basins within the Western Cordillera
between 18 and 22°S (Gaupp et al., 1999; Saez et al., 1999). However, based on oxygen
isotopes, soil morphological characteristics and salt chemistry, Rech et al. (2006) and
Rech et al. (2010) proposed that the onset of hyperaridity could have occurred in the
Atacama Desert earlier, between 19 and 13 Ma. In northern Argentina between 25 and
26°S, recent thermochronologic data from the Puna plateau show that the Eastern
Cordillera was deformed and exhumed at the same time as the Eastern Cordillera further
to the north in southern Bolivia (Carappa et al., 2011). Interestingly, Vandervoort et al.
(1995) also documented an earlier shift from nonevaporitic to evaporitic sedimentary
deposits within Puna plateau basins located between 24-26°S at approximately 15 Ma. A
129
middle Miocene onset of aridity-hyperaridity within the Andean plateau (perhaps as far
south as the Puna) and basins on the western margin of the central Andes would be
consistent with the peak elevations within the Eastern Cordillera that were necessary to
generate sufficient accommodation space for the Caramargo Formation.
Another output of our numerical model during the end-member uplift model
experiment was a time series for sediment leaving the foreland basin (Figures 3 and 4).
Changes in the sediment bypass rates would not be directly recorded in the foreland-basin
stratigraphy, and therefore, one must look at the stratigraphy of adjacent intracratonic
basins or the continental shelf to directly sample this signal. Presently, a topographic
divide exists in southern Bolivia that splits the flow of major rivers draining off the
central Andes north into the Amazon and southeast into the Rio del La Plata Cratonic
Basins. Based on the sediment bypass time series for each of the simulations the sediment
flux component from the central Andes should steadily increase from the Late Oligocene
into the Quaternary with a local minimum between 8-3 Ma. Analysis of drill core
sediments from the Amazon Fan and Ceara Rise show that Andean derived sediments
reached the continental margin between 16.5 and 11.3 Ma (Dobson et al., 2001;
Figueiredo et al., 2009). Between 9 and 6.8 Ma sedimentation rates increased for both the
Amazon Fan and Ceara Rise. This increase in sedimentation rates was interpreted as the
establishment of a transcontinental Amazon drainage system that fully linked the Andean
forelands to the Amazon Fan. Sedimentation rates continued to increase into the
Pliocene-Pleistocene during the period of most rapid uplift within the northern Andes
(Hoorn et al., 1995). The overall increase in sedimentation rates on the Atlantic shelf is
130
predicted by our model. The higher the mean elevation of the mountain belt, the greater
the sediment supply may exceed available accommodation space. Our model also
predicts a factor of two increase in sediment flux leaving the foreland basin during the
Late Pliocene to Pleistocene as foredeep accommodation rates decrease in response to
slower rock uplift rates in the Interandean and Subandean zones. One aspect of our model
results that is not recorded in the Amazon Fan is a Late Miocene to Early Pliocene
decrease in sediment supply rate. The absence of a drop in sedimentation rates may be
expected because a small portion of the Amazon Basin drains the central Andes, and
therefore, sediment supply to the Amazon Fan is more predominantly influenced by the
northern Andes. A stratigraphic data set that would be more predominately influenced by
the central Andes would be the deposits of deepwater fans offshore of the Rio del la Plata
estuary which samples the Andes between 18 and 34°S. To our knowledge no analysis
has been conducted for the Late Tertiary sediments of the Rio del la Plata fans.
A hypothetical eclogite foundering event in the eastern Altiplano region during
the Middle Miocene to Pliocene time was synchronous with increased grain size and
depositional rate in the foreland-basin stratigraphy. Our results show that the distance
from the center of mass of the load is the most important parameter for determining how
the growth and removal of a lower crustal load affected rock uplift and subsidence within
the mountain belt and foreland basin (Figure 6). An inflection point between subsidence
and uplift occurs at a distance of (π/2)α (i.e., approximately 235 km for this study) from
the growing eclogite root load. During the Oligocene to Early Miocene, the foredeep
depozone was located closer to the eclogite root than it would be for the remainder of the
131
simulation (i.e., less than (π/2)α km). As a result, the accumulation of eclogite drove an
additional 100 meters of subsidence, which is more than a factor of 2 greater than
subsidence added to the western and eastern Subandean depositional basins during the
Late Miocene. The greatest deflection occurred directly above the center of the eclogite
root, which was located in the eastern Altiplano and westernmost Eastern Cordillera. This
deflection led to approximately 0.5 to 1 km of rock uplift in the Altiplano. Therefore, an
addition 2 km of surface uplift due to crustal thickening and sediment deposition is
required to achieve the modern mean elevation of the Altiplano if it was located at 1 km
around 10 Ma. Similar amounts of rock uplift due to crustal thickening (i.e., 1-2 km) are
required in the Eastern Cordillera to reach its modern mean and spatially-averaged
maximum elevations. However, less rock uplift due to crustal thickening is necessary if
the average thickness of the eclogite root was significantly larger than the value applied
in this study. We infer that this is less likely because a thicker eclogite root would grow a
significantly sized instability in less than a period of 3 Myr, which is the proposed length
for the delamination period. Our results also show that less than 0.5 km of rock uplift is
contributed to the eastern edge of the Eastern Cordillera and western Interandean zones
by eclogite removal. Barke and Lamb (2006) calculated that the San Juan del Oro
paleosurface that overlies the eastern part of the Eastern Cordillera was uniformally
uplifted by approximately 1 km. Therefore, the eclogite root must be located closer or
beneath to the forethrust of the Eastern Cordillera to uniformly uplift it by 1 km.
However, tomography data suggests that foundering is more likely beneath the Altiplano
132
and western part of the Eastern Cordillera (Beck and Zandt, 2002). Thus, part of the 1 km
of rock uplift of the San Juan del Oro surface is likely the result of crustal thickening.
Climate change was the final process that we tested for the foreland basin of the
central Andes of southern Bolivia. Erosion rates and transport coefficients between 43
and 9 Ma were a factor of 2 less than the values between 9 and 0 Ma. As a direct result
peak sediment flux into the foreland basin and sediment bypass rates were significantly
less than the values resulting from the rapid uplift model (Figure 8). At the onset of the
South American monsoon, both sediment bypass rates and sediment flux rates into the
foredeep increased significantly. The increase in sediment supply to the foredeep is
predominantly controlled by the rapid uplift of the mountain belt instead of by the
increase in mean annual precipitation. Increasing erosion rates and decreasing transport
rates should lead to steeper slopes in the proximal foreland basin. Both sediment supply
and transport capability increased as mean annual precipitation increased, which lead to
minimal change in the regional slopes as these effects cancel each other. Regardless,
regional slopes in the model appear to be predominately affected by rapid subsidence due
to crustal thickening near the deformation front instead of by increasing precipitation.
Another way to gauge the effect of climate change is to track the boundary between
gravel and sand deposition. Paola et al. (1992) demonstrated that sinusoidal variation in
both sediment supply and transport coefficients lead to migration of the gravel-sand
interface especially if the forcing period is small compared to the basin equilibrium time.
Our results show that the first order migration of the gravel-sand boundary appears to be
unaffected by the factor of 2 increase in both of these parameters over the timescales that
133
we are sampling (Figure 9). Based on our results the threshold slope term appears to be a
more primary control on gravel progradation than the transport coefficient term. Rock
uplift in the hinterland leads to steeper slopes that are capable of transporting gravel. As a
result, gravel trapped near the mountain front can rapidly prograde toward the new
deformation front when it is located in a wedge top basin. Eventually, rock uplift leads to
the exhumation of pre-Tertiary units near the deformation front that can produce gravel
during bedrock incision. Conversely, regional slope within the foreland is insufficient to
transport gravels far into the foredeep. Therefore, a combination of processes causes
progradation of the gravel-sand interface during forward propagation of the deformation
front; climate change appears to have played a secondary role in controlling gravel
progradation for the central Andes.
6. Conclusions
Based on our modeling results, we propose that the early surface uplift history
(i.e., prior to 22 Ma) of the eastern margin of the central Andes more closely resembled
the gradual uplift model. When the rigidity of the South American plate ranged between
1.5x1023 and 4.0x1024 Nm, surface uplift of the core of the Eastern Cordillera in the rapid
uplift model produced grossly under-matched sediment accommodation during the
deposition of the >2km thick Camargo Formation. The gradual uplift model more closely
fits the observed sediment thicknesses for the Camargo Formation located in the Eastern
Cordillera, and thus, the core of the Eastern Cordillera experienced rock uplift rates that
would have lead to significant topography (i.e., peak elevations near 2-3 km) by the early
Miocene if basin-averaged erosion rates were on the order of 10-4 m/yr. Further crustal
134
thickening during the middle Miocene may have increased mean topography of the
Eastern Cordillera to the point where it became an effective barrier to easterly moisture
derived from the Atlantic; in turn, this initiated aridity in the Western Cordillera and
Atacama Desert regions at that time. During the Late Miocene, the Interandean and
western Subandean zones likely underwent rapid rock and surface uplift to produce the
amount of accommodation required to store the thick Tariquía Formation which was
deposited within a period of 2 million years.
Our results also support the hypothesis that the first-order trends in the Tertiary
Subandean zone foreland-basin stratigraphy are predominantly influenced by its distance
from the approaching mountain belt. An increase in topographic loads located less then
(3/4)πα km from the a depozone can cause deflections up to the order of a km. Beyond
this distance deflection ranges up to 100s of meters near the forebulge. During the Late
Miocene the eclogite root was located more than a distance of (3/4)πα from the
Subandean depozone. As such, the accommodation rates within the central-eastern
Subandean zone foredeep were positive, but only on the order of 10s meters. Therefore,
the deflection caused by an approaching mountain belt exceeded the deflection caused by
eclogite delamination. Also based on our results, the factor-of-five increase in
depositional rates was not likely the result of an increase in erosion rates and sediment
transport rates caused by climate change. Increased erosion rates would lead to isostatic
rebound of the mountain front, and thus decreased sediment accommodation within the
foredeep, if crustal thickening does not increase as well. If crustal thickening does
increase with erosion rates as in our model, then the additional sediment supplied
135
bypasses the foreland basin because it is already overfilled. In addition to accommodation
rates, the location of the deformation front in time appears to exert a primary control on
the gravel-sand interface. Increasing erosion rates and sediment transport rates by a factor
of 2 due to the onset of the South American Monsoon at 9 Ma did not generate a longterm progradation in the gravel-sand interface. Instead, the gravel front retreated toward
the deformation front due to the rapid subsidence rates within the foredeep basin.
Actively uplifting fold-and-thrust belts are able to achieve channel slopes that are above
the threshold for gravel entrainment and produce gravel by bedrock incision. As a direct
result, the gravel-sand interface rapidly progrades when the deformation front propagates
into the foreland basin.
Acknowledgements
Our research was supported in part by ExxonMobil funding. We thank Peter
DeCelles and Joellen Russell for their helpful comments.
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Tables and Figures:
Table 1: Parameters used in end-member model simulations
ke (m/yr)
kg (m2/yr)
ks (m2/yr)
φ
α (km)
ρs (kg/m3)
ρm (kg/m3)
ρe (kg/m3)
g (m/s2)
1.0x10-6
1.0x104
6.5x104
0.001
150
2750
3300
3600
9.8
Figure 1: Digital Elevation model for the central Andes displaying the tectonomorphic
regions after Barnes et al. (2008). Topography is from the SRTM 90 m data set. The
tectonomorphic regions are the following: WC, Western Cordillera; AL, Altiplano; EC,
Eastern Cordillera; IA, Interandean zone; SA, Subandean zone. The thick white lines are
thrust faults located on the boundary between major divisions and the thin white lines
mark political boundaries.
149
Figure 2: Generalized stratigraphic columns for the Cenozoic stratigraphy of the Chaco
foreland basin after DeCelles and Horton (2003), Uba et al. (2006) and Uba et al. (2007).
Formation thicknesses are displayed beneath the formation names. The questions marks
150
represent uncertainty in the boundaries between formations due to a lack of absolute age
data. A detailed discussion of the age uncertainties for the Cenozoic stratigraphy of the
Eastern Cordillera, Interandean and Subandean zones can be found in DeCelles and
Horton (2003) and Uba et al. (2006).
Figure 3: (A) Topographic evolution of the central Andes and (B) time series of sediment
supply rates into and out of the foreland basin for the gradual uplift model. The thick
black lines in Figure (A) represent snapshots of topography throughout the simulation
and the thick gray lines represent the maximum and mean topography for a north-south
sweep of topography between 18-21°S. The black rectangles represent the location of the
forebulge crest during each snapshot in time with the oldest forebulge location on the left.
In Figure (B), the thick line represents the sediment flux leaving the backbulge basin or
right edge of the model and the thin lines represent the sediment flux at the deformation
front into the foredeep of the foreland basin.
151
Figure 4: (A) Topographic evolution of the central Andes and (B) time series of sediment
supply rates into and out of the foreland basin for the rapid uplift model. The thick black
lines in figure (A) represent snapshots of topography throughout the simulation and the
thick gray lines represent the maximum and mean topography for a north-south sweep of
topography between 18-21°S. The black boxes represent the location of the forebulge
crest during each snapshot in time with the oldest forebulge location on the left. In figure
(B), the thick line represents the sediment flux leaving the backbulge basin or right edge
of the model and the thin lines represent the sediment flux at the deformation front into
the foredeep of the foreland basin.
152
Figure 5: Uncompacted Tertiary sediment thickness for depozones located in the eastern
Subandean zone (ESA), western Subandean zone (WSA), western Interandean zone (IA)
and within the Eastern Cordillera forethrust region (EC) for the (A) gradual uplift and (B)
rapid uplift models. The dotted, solid and dashed lines represent simulations with flexural
parameters of 100, 150 and 235 km. Shaded regions represent the times when the
Tariquía Formation of the Subandean zone and Camargo Formation of the Eastern
Cordillera and Interandean zone were deposited. The subsidence curve from Uba et al.
(2006) for the location in the Subandes that we sampled is represented by black triangles
in both plots.
153
Figure 6: Topographic and flexural profile snapshots for the eclogite delamination model
at (A) 25 and (B) 10 Ma. The shaded gray regions represent Tertiary sedimentary
deposits and the pre-Tertiary bedrock is white. Lithospheric deflection caused by
accumulation and delamination of an eclogite load is represented by a thick black line,
whose scale is show on the right vertical axis, and the position of the eclogite load is
represented by the black horizontal bars.
Figure 7: (A) Time series data for sediment fluxes into and out of the foreland basin and
(B) uncompacted sediment thickness for depozones located in the eastern Subandean
(EA), western Subandean (WSA), Interandean (IA) and Eastern Cordillera (EC)
tectonomorphic zones for the eclogite delamination model. The black symbol at the top
of figure A and the bottom of figure B represents the growth and delamination of the
eclogite root between 25 and 10 Ma. In figures A and B the thick lines represent the
eclogite delamination model and the thin dashed lines represent the rapid uplift model.
154
Figure 8: Sediment flux time series data for the climate change model. The thick line
represents the sediment flux leaving the backbulge basin during the climate change model
and the dashed lines represent the sediment flux leaving the backbulge basin during the
rapid uplift model. The dashed vertical line represents the onset of the South American
monsoon.
Figure 9: Gravel-sand interface time series data for the (A) gradual and rapid uplift
models and the (B) rapid uplift and climate change models. In figure A the gradual and
155
rapid uplift models are represented by the solid and dotted lines respectively and in figure
B the climate change and rapid models are represented by the solid and dotted lines
respectively. The location of the deformation front is shown by the dashed line in both
plots.
156
APPENDIX C: QUANTIFYING THE EFFECT OF HYDROLOGIC
VARIABILITY ON BEDLOAD SEDIMENT TRANSPORT IN
ALLUVIAL CHANNELS
Manuscript in review with Geomorphology
Todd M. Engelder, Department of Geosciences, University of Arizona, 1040 E. Fourth
St., Tucson AZ, 85721, USA.
Jon D. Pelletier, Department of Geosciences, University of Arizona, 1040 E. Fourth St.,
Tucson AZ, 85721, USA.
Abstract
Existing equations for predicting the long-term bedload sediment flux in alluvial channels
include mean discharge as a controlling variable but do not explicitly include variations
in discharge through time. In this paper, we develop an analytic equation for the longterm bedload sediment flux in alluvial channels that incorporates instantaneous sediment
transport equations and the frequency-size distribution of flood events, taking into
account both the mean and coefficient of variation of discharge for a channel with a
prescribed slope and grain size. Three applications of the resulting equation are
considered. First, the equation is used to refine estimates for the effective diffusivity of
alluvial channels within the framework of the diffusion model for longitudinal profile
evolution. Second, the geomorphic effectiveness of end-member annual-runoff regimes is
considered, taking into account the inverse relationship between the mean and coefficient
of variation of discharge in alluvial channels. The results indicate that an alluvial channel
with high effective annual runoff transports more sediment per unit time than the same
157
channel would with low effective annual runoff despite the fact that high effective annual
runoff conditions may lack the extreme events characteristic of low effective annual
runoff conditions. Third, the return period of the effective discharge (i.e. the discharge
that contributes the most sediment to the long-term bedload sediment flux) is calculated
as a function of climate. Channels in humid-to-temperate climates have effectivedischarge return periods of months to decades, while the return periods of the effective
discharge for extremely arid climates can be up to several hundred years.
Index Terms- sediment transport, climate impacts, stochastic hydrology, diffusion
158
1. Introduction
A better understanding of long-term bedload sediment fluxes is needed to properly
manage floodplains and reservoirs and to predict how alluvial channels will respond to
future climatic and hydrologic changes (Harlin, 1978; Goodwin, 2004). Long-term in this
context refers to time scales of decades to centuries, i.e. sufficiently long that the
estimated sediment flux for a given location includes the cumulative effects of many
flood events but not so long that the estimate averages over the effects of different
climatic conditions. Quantifying mean bedload sediment fluxes over geologic time scales
is necessary for understanding the evolution of sedimentary basins, because upstream
sediment flux acts as a principal boundary condition for terrestrial and marine depozones
(Paola et al., 1992; Molnar, 2004).
Here we derive an analytic equation for the long-term sediment transport for an alluvial
channel with a prescribed grain size, slope, and mean and coefficient of variation of
discharge. Wolman and Miller (1960) first proposed that the geomorphic work (i.e. longterm sediment transport) performed by a channel could be quantified by integrating the
product of a sediment transport function and a frequency-size distribution of discharge
over the range of all possible discharges. They referred to the product of these two
functions as the “effectiveness function.” An abundant literature now exists that
computes the long-term sediment flux by integrating the product of the frequency
distribution of daily discharge with a site-calibrated sediment rating curve for specific
channels (e.g. Kettner and Syvitski, 2008), but no generally-applicable analytic equations
159
for predicting long-term sediment flux are available that incorporate both the mean and
coefficient of variation of discharge. Such an equation would be useful for predicting the
rates of topographic evolution of alluvial channels as well as for predicting channel
response to climatic and anthropogenic changes.
Sediment discharge can be divided into a suspended-load component, i.e. sediment that
moves in the water column while only occasionally touching the bed, and a bedload
component, i.e. sediment that moves close to the bed by repeated instances of rolling,
sliding, and saltation. In this paper we focus on the bedload component for three reasons.
First, bedload sediment transport is more closely related to local hydraulic conditions.
Second, many bedload formulae have been proposed that work reasonably well. Third,
the topographic evolution of alluvial channels is most closely related to bedload because
suspended sediment load is sourced primarily from hillslopes and, while some deposition
of suspended load can occur in channel beds, most suspended sediment moves from
hillslopes to depositional basins (alluvial fans and deltas) without affecting the channel
geometry to the same extent as bedload. In contrast, the variation in bedload sediment
fluxes along a channel longitudinal profile is the primary driver for aggradation and
incision of alluvial channel beds.
The diffusion equation is a commonly used model for simulating the topographic
evolution of alluvial channels in sedimentary basins over geologic timescales (Flemings
and Jordan, 1989; Paola et al., 1992; Marr et al., 2000). The diffusion equation applies
160
mass-conservation principles to predict the rate of erosion or deposition at a point along a
channel to the gradient of long-term sediment flux at that point. As such, proper
application of the diffusion equation requires accurate models for how long-term bedload
sediment fluxes relate to the controlling parameters of channel slope, grain size, and
hydrologic forcing. More specifically, the diffusion equation assumes that sediment
fluxes are proportional to the local slope gradient. It is written as the following:
∂ 2h
∂h
=κ 2
∂t
∂x
(1)
where h is the elevation of the channel bed (m), κ is diffusivity (m2/yr), t is time (yr) and
x is the horizontal distance with respect to the model origin (m). The diffusivity
parameter of the diffusion equation, which is the proportionality constant between
sediment flux and channel slope, is a function of all of the variables that control bedload
sediment flux, i.e. grain size, channel width, and the hydrological forcing (i.e. the mean
and variability of discharge). Paola et al. (1992) derived an expression for diffusivity
using the Meyer-Peter and Mueller (1948) sediment transport relation. Paola et al.’s
(1992) expression was a function of mean discharge and channel morphology. As such,
it did not incorporate grain size or the variability of discharge. Later, Marr et al. (2000)
derived a diffusivity that included grain-size effects and a term for hydrologic variability;
however, the authors chose to set their variability parameter to unity to simplify the
results. There is clear evidence, however, that the variability of discharge affects long-
161
term sediment transport rates. For example, studies that carefully document the temporal
variability of sediment transport often show that the largest events transport the most
sediment (Inman and Jenkins, 1999). As such, the mean discharge cannot be the only
variable that controls long-term sediment flux. At larger spatial and temporal scales,
Leier et al. (2005) documented lower slopes of fluvial megafans in areas of higher
discharge variability but similar mean discharge worldwide. Leier et al.’s (2005)
observation implies that variability in discharge must be a controlling factor on long-term
sediment transport rates because the slopes of the fans reflect the effective diffusivity of
the fluvial system. In this paper we derive an analytic expression for the diffusivity that
incorporates hydrologic variability.
One explanation why discharge variability is important for long-term transport rates is
that sediment transport is a nonlinear function of discharge. Analysis of bedload transport
in gravel-bed channels reveals that most power-law rating-curve exponents range
between 1.5 and 4.4 for gravel-bed channels (Goodwin, 2004; King et al., 2004).
However, power-law rating-curve exponents can also exceed 5.0 when armoring occurs
(Emmett and Wolman, 2001). Such strongly nonlinear behavior increases the importance
of rare, large flood events relative to more frequent, smaller floods.
Molnar (2001) posed the question of whether channels in arid or humid climates transport
more sediment as bedload. Analysis of channel hydrographs reveals that discharge
variability, expressed as a coefficient of variation, increases with increasing aridity.
162
Turcotte and Greene (1993), for example, analyzed peak annual discharge for 10
channels in various climates across the United States. Their results showed that the ratio
of the frequency of the largest floods to that of the mean flood increased significantly
with increasing aridity. Molnar et al. (2006) found a similar correlation between the
power-law exponents for flood frequency and effective annual precipitation for an
additional 159 channels across the United States. In the most comprehensive analysis to
date, McMahon et al. (2007) analyzed the sample statistics of 1221 channels globally.
Their results confirm a negative correlation between the coefficient of variation (i.e., the
standard deviation normalized by the mean) of annual discharge and the mean annual
runoff. Therefore, as climates shift to greater aridity, the variability in discharge
increases.
Molnar (2001) concluded that channels in arid climates would transport more sediment
than the same channel in a humid climate. In effect, Molnar (2001) argued that the
increase in the flashiness of discharge more than compensated for the decrease in mean
discharge with increasing aridity. The reason for this is related to the fact that large floods
are so much more effective, per unit discharge, than small floods. In a revised analysis,
however, Molnar et al. (2006) pointed to a normalization error in Molnar (2001) and
stated that unless the threshold for entrainment was high enough to be exceeded only by
very large floods, greater aridity would not likely result in larger transport rates compared
to more humid conditions. Molnar et al.’s (2006) analysis was limited, however, in that
they did not explicitly incorporate sediment transport into their study. As such, the
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relationship between climate and long-term sediment transport is still not well
constrained and the question posed by Molnar (2001) remains to be fully answered.
In addition to diffusivity, our analytic solution for the long-term bedload sediment
transport also constrains the effective discharge. Wolman and Miller (1960) defined the
effective discharge as the discharge responsible for transporting more sediment than any
other discharge. Several studies provide analytic solutions for the effective discharge
(Nash, 1994; Goodwin, 2004; Doyle et al., 2007). Nash (1994), for example, showed that
the effective discharge for suspended sediment was a function of both the mean and
coefficient of variation of discharge when the effectiveness function is the product of a
sediment rating curve and a two-parameter lognormal distribution. Goodwin (2004)
extended this analysis to include the use of effectiveness functions that were the product
of more general sediment transport equations and several types of frequency
distributions. Although transport thresholds were included in deriving some of his
analytic solutions, results showed that the threshold for transport did not change the
magnitude of the effective discharge as long as the effective discharge was significantly
greater than the threshold for entrainment. These studies and others show that both
numerical and analytic solutions for effective discharge can be used to accurately predict
the effective discharge for bedload sediment transport in modern channels (Torizzo and
Pitlick, 2004; Barry et al., 2008). Goodwin (2004) also stated that the analytic solutions
he derived could be used to predict changes in effective discharge caused by future
changes in hydrology and sediment supply. However, these studies did not take into
164
consideration the covariation between the mean and the coefficient of variation of
discharge considered in this paper.
Here we derive an analytic equation for the long-term bedload sediment flux that includes
the mean and coefficient of variation of discharge. We then explore three applications of
that equation. First, recasting the analytic solution as a function of channel gradient yields
an analytic solution for the diffusivity as a function of effective mean discharge and the
coefficient of variation of mean daily discharge. We then use this equation to determine
the sensitivity of diffusivity to each controlling parameter and compare our results to
reported diffusivity values from the literature. Second, we determine the relationship
between long-term bedload sediment flux and hydrologic distribution, which is a function
of climate, for an alluvial channel of a prescribed slope and grain size. Third, we derive
an expression for the effective discharge and the return period of that effective discharge.
The results of that analysis enable us to assess how the magnitude and frequency of the
channel-forming discharge depends on the hydrologic distribution and, by extension, to
climate.
2. Analytic Solutions
2.1. Derivation of an analytic equation for long-term bedload sediment flux
165
The long-term bedload sediment flux per unit channel width in an alluvial channel is
given by the integral of the effectiveness function. Although the effectiveness function is
generally cast in terms of volumetric discharge, here we choose to write it as a function
of Shields stress, τ* (i.e. the dimensionless shear stress), in order to simplify the
expression and facilitate comparison between channels of different geometries and
textures. The expression for the long-term bedload sediment flux as a function of Shields
stress and critical Shields stress is:
∞
q s * = ∫ qs * (τ * , τ *cr ) f (τ* )d τ *
τ *cr
(2)
where qs* is the instantaneous, non-dimensional volumetric bedload sediment flux per
unit channel width, and τ*cr is the critical Shields stress for sediment entrainment (also
dimensionless). In this paper, all variables that have a bar above them represent timeaveraged (i.e., long-term) values.
Sand- and gravel-bed channels must be considered separately because different sediment
transport formulae apply to each type of channel (discussed in the next paragraph). For
sand-bed channels, we set the lower limit of integration in (2) to 0 and the upper limit to
∞. The lower limit of integration is 0 because commonly-used sediment transport
formulae for sand-bed channels, including the formula we adopt in this paper, do not
include a threshold for transport. At the upper end of the integration, a natural upper
bound to Shields stress exists that is largely controlled by the limit of water that can
166
delivered by a storm to a basin over a given period of time. Instead of solving for the
upper bound as a function of basin parameters (e.g. size and drainage density) and
climate the upper bound is set to ∞. The simplification of choosing no upper limit for the
range of integration is reasonable because the frequencies of discharges located at the
extreme end of the distribution tail are sufficiently low that such storms comprise a very
small component of the long-term sediment transport rate. For gravel-bed channels, we
set the lower limit of integration to equal the critical Shields stress because, by definition,
no transport occurs below the critical Shields stress. However, two possible choices exist
for the upper limit in the gravel-bed channel case: (1) Shields stress can continue to
increase as discharge increases (necessitating an upper limit of integration of ∞), or (2)
Shields stress can be limited to values between 1.2 and 1.4 times the critical value to
account for channel widening in response to large floods (i.e., the self-forming channel of
Parker (1978)). If the channel is self-forming, then the integral must be the sum of two
terms: an integral similar to (1) when the Shields stress is less than either 1.2 or 1.4 times
the critical Shields stress and an integral with a constant transport rate when the Shields
stress exceeds either 1.2 or 1.4 times the critical Shields stress. Like gravel-bed channels,
sand-bed channels do widen in response to floods that generate shear stresses sufficient to
entrain channel bank sediment. However, we chose to not to explore the effects of
limiting shear stress in sand-bed channels because the specific value for the upper bound
in Shields stress is not as well constrained in the literature for sand-bed channels.
167
Two general classes of sediment transport formulae can be applied to solve for the longterm bedload sediment flux: site-specific sediment rating curves and general transport
equations. Sediment rating curves are generally cast as a power-law function of discharge
and the coefficient and exponent of each rating curve are calibrated to a specific reach
within a channel and for a given period of time. Rating curves have the advantage of a
simple functional form and they match observed sediment fluxes in natural streams better
than general transport equations. However, they have the disadvantage of requiring sitespecific and time-specific calibration. More general transport equations are required to
predict instantaneous sediment fluxes over a broad range of channel conditions without
site-specific calibration. Numerous sediment transport formulae have been developed
over the last century for this purpose. Empirical equations that predict sediment flux as a
function of shear stress (Meyer-Peter and Muller, 1948; Engelund and Hansen,1967;
Parker et al., 1982; Wiberg and Smith, 1989), as a stochastic function of sediment
movement (Einstein, 1950), or as a function of stream power (Bagnold; 1980) have been
tested against observed sediment transport rates for channels (e.g. Gomez and Church,
1989; Barry et al., 2004). Gomez and Church (1989) found that the majority of bedload
sediment transport formulae over-predict transport rates. These authors attributed this
over-prediction to bed-surface armoring and insufficient sediment supply to the reach.
More recently, Barry et al. (2004) tested bedload sediment transport equations against
sediment transport data collected in gravel-bed channels by the USGS and USDA Forest
Service in Idaho and found no consistent relationship between sediment transport
formulae performance and the degree to which each formula was calibrated or to the
168
complexity of the equation. While bedload sediment transport formulae have a limited
ability to accurately predict instantaneous fluxes within channels, the averaging effect of
computing long-term sediment fluxes mitigates these problems to some extent. Both
studies also found that formulae with thresholds for entrainment often erroneously
predicted zero transport when the discharge was finite but below the predicted threshold
for transport. As a result, Barry et al. (2004) found that the Ackers and White (1973) and
Parker et al. (1982) equations preformed better than equations that contain a threshold
term for entrainment (e.g. Meyer-Peter and Muller (1948)). Despite the increase in
accuracy, however, the complexity of the Ackers and White (1973) and Parker et al.
(1982) equations makes both integration and calibration of the analytic solutions difficult.
Therefore, we compute bedload sediment flux in gravel-bed channels in this study using
the following relationship derived by Wiberg and Smith (1989):
q s * = 9.64 τ *1 / 6 (τ * − τ *cr ) 3 / 2
(3)
Equation (3) is applicable over a broad range of grain sizes from medium sand to gravel
(i.e., 0.35 to 28.6 mm). For sand-dominated channels with grain sizes less than 2 mm, we
use the Engelund and Hansen (1967) transport equation, i.e.
⎛ 0.05 ⎞ 5 / 2
⎟⎟ τ *
q s * = ⎜⎜
⎝ Cf ⎠
(4)
169
where Cf is a dimensionless drag coefficient, assumed here to be 0.01. Equation (4) does
not contain a threshold for entrainment and is a simple power-law function of Shields
stress.
Analyses of hydrologic time-series data show that both mean daily and mean annual
channel discharge can follow normal, lognormal, power-law, and other distributions with
parameters that depend on drainage basin area and climate (e.g. Leopold et al., 1964;
Turcotte and Greene, 1993; Molnar et al., 2006). Further analysis of channels from North
America and Europe reveal that the two-parameter lognormal and gamma distributions
best describe the time series data for mean annual discharge (Goodwin, 2004; McMahon
et al., 2007). Equation (2) is cast as a function of Shields stress instead of discharge, and
thus, a representative distribution for Shields stress is required for this analysis. When
channel flow is uniform and steady, Shields stress is a power-law function of discharge.
In this paper we assume that discharge, and hence Shields stress, follows a two-parameter
lognormal distribution. We use the lognormal distribution because it is a commonly used
two-parameter distribution for discharge that can accurately represent the entire
distribution of discharge events including small, common discharges and large, rare
discharges (Goodwin, 2004; McMahon et al., 2007). Written in terms of τ*, this
distribution is given by
(
⎛
⎞
⎛ − lnτ * − μ lnτ
1
*
⎟ exp⎜
f (τ * ) = ⎜
2
⎜
⎜ τ * σ lnτ 2π ⎟
2 σ lnτ*
⎝
*
⎝
⎠
)
2
⎞
⎟
⎟
⎠
(5)
170
where σlnτ* is the standard deviation of ln τ* and μlnτ* is the mean of ln τ*. Substituting (3)
and (5) into (2) yields an effectiveness function for τ* that can be applied to gravel-bed
channels in cases when τ* is not limited near τ*cr for high discharges:
∞
qs* =
∫
⎛
1/ 6
9.64 τ * (τ * − τ *cr ) 3 / 2 ⎜
τ*cr
1
⎜ τ * σ lnτ
*
⎝
(
⎛ − lnτ * − μ lnτ
⎞
*
⎟ exp⎜
2
⎜
⎟
2π ⎠
2 σ lnτ*
⎝
)2 ⎞⎟
⎟
⎠
d τ*
(6)
A similar expression for sand-bed channels can be obtained by substituting (4) and (5)
into (2). However, the limits of integration are between 0 and ∞ for the sand-bed case. In
order to compute the integral of (6) analytically, it is necessary to perform a Taylor
expansion of the Wiberg and Smith (1989) formula, i.e. (3), in order to yield an infinite
series of solvable integrals. We used the following Taylor expansion:
⎛3⎞
⎛3⎞
(τ * − τ *cr ) 3 / 2 = τ * 3 / 2 − ⎜ ⎟ τ *1 / 2 τ *cr + ⎜ ⎟ τ * −1 / 2 τ *cr 2 + ....... + H .O.T .
⎝2⎠
⎝8⎠
(7)
where H.O.T. stands for higher-order terms. Upon integration of (6), the resulting
equation is a function of the mean and standard deviation of the normally distributed ln
τ*. As such, we must solve for the mean and standard deviation of a lognormally
transformed variable in terms of the mean and variance of the original variable, which
yields the following solutions:
171
μ lnτ*
(
⎛ ln 1 + C v,τ 2
*
= ln τ * − ⎜
⎜
2
⎝
( )
(
σ lnτ* = ln 1 + C v,τ*
where τ
*
)⎞⎟
(8)
⎟
⎠
)
2 1/ 2
(9)
is the mean of τ*, Cv,τ* is the coefficient of variation of τ*. Substituting (8) and
(9) into the infinite-series solution for the integral of (6) results in the following
expression:
∞
qs * = ∑ α τ
k =1
⎛ 3 k −8 ⎞
−⎜
⎟
⎝ 3 ⎠
*
(1 + C )
2
v, τ*
⎛ ( 3 k −8 )( 3 k −5 ) ⎞
⎜
⎟
18
⎝
⎠
[1 − erf (β )]
13
⎛
2
⎜ − 3 ln τ + (3k − ) ln(1 + C v,τ* ) + 3 ln τ* cr
*
2
β=⎜
2
⎜
3 2 ln(1 + C v,τ* ) (1 / 2)
⎜
⎝
⎧ ⎛⎜ 9.64 ⎞⎟
⎪⎝ 2 ⎠
α = ⎨ ⎛ 9.64 ⎞⎛⎜ ( −1) k −1 τ *kcr−1 ⎞⎟ k −1 ⎛ 5 ⎞
⎪ ⎜⎝ 2 ⎟⎠⎜ (k −1)! ⎟ mπ=1⎜⎝ 2 − m ⎟⎠
⎠
⎝
⎩
(10)
⎞
⎟
⎟
⎟
⎟
⎠
if , k =1
if , k >1
where k and m are summation indices and α and β are constants defined in equation 10.
The symbol erf() stands for the Error function. Equation 10 predicts the long-term
bedload sediment flux for a given lognormal distribution of τ*. In practice, only the first
few terms in the series are required in order to implement (10) because the series
172
converges to several-decimal-point-accuracy with just a few terms for most values of
τ and Cv,τ*. We verified the accuracy of the form of (10) by comparing its predictions to
*
a numerically-integrated calculation of (6). This validation procedure illustrates the
power of a closed-form analytic expression for long-term sediment flux. Although the
expression in (10) is lengthy, numerical integration requires different integration step
sizes for different values of Cv,τ* (to insure that the final solutions for long-term sediment
flux represent the entire range of contributing discharge), and thus determining the
relationships between long-term sediment flux and τ and Cv,τ* is less than
*
straightforward. The series solution, in contrast, provides an exact result without
numerical integration.
For gravel-bed channels that adjust to high Shields stresses by widening, the integral is
divided into two ranges of Shields stress:
∞
qs * = ∑ α τ
k =1
⎛ 3 k −8 ⎞
−⎜
⎟
⎝ 3 ⎠
*
(1 + C )
γ = 9.64[(1 + ε ) τ *cr ]
2
v, τ*
1/ 6
(ε τ *cr )
⎛ ( 3 k −8 )( 3 k −5 ) ⎞
⎜
⎟
18
⎝
⎠
3/ 2
[erf (β ) − erf (β )] + γ
(11)
⎡
⎛ ln((1 + ε ) τ *cr ) − ln τ + (1 / 2) ln(1 + C v,τ 2 ) ⎞⎤
*
⎟⎥
*
⎢ 1 − 1 erf ⎜
⎟⎥
2 1/ 4
⎢ 2 2 ⎜⎜
2 ln(1 + C v,τ* )
⎟
⎢⎣
⎝
⎠⎥⎦
173
where γ is the contribution to the long-term sediment flux when Shields stress is greater
than (1+ε)τ*cr and ε is equal to a constant between 0.2 and 0.4. The terms α and β are the
same as in (10) except that τ*cr is replaced with (1+ε)τ*cr in the left-most erf (β) term.
We also solved for the long-term bedload sediment flux by integrating the product of the
Engelund and Hansen (1967) transport equation and the lognormal distribution. In this
case, performing a Taylor expansion on the transport equation was not necessary because
the resulting integral has a known solution. The following is the expression for long-term
bedload sediment flux applicable to sand-bed channels:
(
⎛ 0.05 ⎞ 5 / 2
⎟⎟ τ * 1 + C v,τ* 2
q s * = ⎜⎜
⎝ 2C f ⎠
)
15 / 8
(12)
In order to apply (10-12) to a specific set of climatic and geomorphic conditions we must
relate the mean channel parameters (i.e., grain size, grain density, channel bed slope and
the frequency-size distribution of discharge) to the Shields stress. When discharge is
relatively steady and uniform, the instantaneous discharge is related to the instantaneous
τ* through the following equations:
τ =
*
(Qn )3 / 5 S 7 / 10
⎛ ρ − ρf
w3 / 5 ⎜ s
⎜ ρf
⎝
⎞
⎟d
⎟
⎠
(13)
174
Q 3 / 10 n 3 / 5 S 7 / 10
τ =
*
⎛ ρ − ρf ⎞
⎟d
b3/ 5 ⎜ s
⎜ ρf ⎟
⎝
⎠
(14)
where Q is the volumetric discharge per unit channel width (m3/s), n is the Manning’s
roughness coefficient (approximately 0.035 for moderately rough gravel-bed channels), S
is the local channel bed slope, d is the sediment grain diameter (m), w is the channel
width (m), ρs is the grain density (kg/m3), ρf is the fluid density (kg/m3) and b is the
proportionality coefficient for the power-law relationship between w and Q (i.e., its units
are s1/2/m1/2 when Q has units of m3/s and the power-law exponent is equal to 1/2). We
solved for a relationship between Shields stress and volumetric discharge because using
volumetric discharge facilitates comparison with data reported in the literature. A squareroot relationship between channel width and discharge (i.e., w = bQ1/2) was applied to
convert volumetric discharge to discharge per unit channel width to yield (14). Several
studies have constrained the range of values for b and the power-law exponent relating
width and discharge for U.S. channels located in humid to semi-arid regions (Blench,
1952; Leopold and Maddock, 1953; Simons and Albertson, 1963). They found that b
values vary from approximately 3.1 to 6.3 depending on the erodibility of the bank
material. The exponent relating width and discharge ranged between 0.50 and 0.51. The
instantaneous Shields stress is also a function of grain diameter d. Of course, channels are
not comprised of sediment grains of uniform diameter. As such, the value of d should be
chosen to be a representative value of the channel bed material, e.g. d50 or d80 (i.e., the
175
median or the 80th percentile grain size of the distribution, respectively). Our analytic
approach, however, is not limited to channel beds with uniform grain sizes or
distributions that can be characterized by a single grain size. In the case of more
complicated grain size distributions, the range of channel bed grain size can be divided
into multiple bins. Then our analytic solution can be used to compute the component (i.e.,
normalized by the bin’s weight percent of the distribution) of long-term bedload sediment
transport rates contributed by each grain size bin to the total long-term bedload sediment
transport rate. Although long-term bedload sediment transport rates can be calculated for
a fixed distribution of grain sizes, long-term changes in channel texture due to hydraulic
sorting and changes in upstream sediment supply cannot be properly addressed by our
analytic solution. As such, sensitivity study results discussed later in the paper are
presented for channels of uniform grain size.
Long-term sediment fluxes result from the cumulative effect of many discharge events.
As such, equations (13 and 14) were modified to relate the mean and variation of Shields
stress to those of discharge. We derived these equations by first linearizing (13) and (14)
as functions of ln Q through a logarithmic transformation. Then, we replaced the
logarithms of Shields stress and discharge with the mean and standard deviations of the
logarithms of Shields stress and discharge. This was accomplished by using standard
relationships for the effect of linear transformation on the mean and the variance of a
variable (i.e., ln τ* is a linear transform of ln Q). As an example, we show the equations
that were derived from (14):
176
μ lnτn
⎛
⎜
⎜ n 3 / 5 S 7 / 10
= (3 / 10) μ lnQ + ln⎜
⎜ 3 / 5 ⎛ ρs − ρ f
⎜⎜ b ⎜⎜
⎝ ρf
⎝
⎞
⎟
⎟
⎟
⎞ ⎟
⎟d ⎟
⎟ ⎟
⎠ ⎠
σ lnτn = (3 / 10) σ lnQ
(15)
(16)
where μlnQ and σlnQ are the mean and the standard deviation of ln Q. Equations similar to
(8) and (9), but written in terms of discharge instead of Shields stress, were then
substituted into (15) and (16) to solve for the relationship between the mean and variation
of both Q and τ*:
⎛
⎜
⎜
Q 3 / 5 n 3 / 5 S 7 / 10
τ =⎜
* ⎜
⎛ ρs − ρ f ⎞
3/5
⎟d 1 + C v,Q 2
⎜⎜ w ⎜⎜
⎟
⎝ ρf ⎠
⎝
(
⎛
⎜
⎜
Q 3 / 10 n 3 / 5 S 7 / 10
τ =⎜
* ⎜
⎛ ρs − ρ f ⎞
3/ 5
⎟d 1 + C v,Q 2
⎜⎜ b ⎜⎜
⎟
⎝ ρf ⎠
⎝
(
[(
C v,τ* = 1 + C v,Q
)
2 9 / 25
]
− 1 1/ 2
⎞
⎟
⎟
⎟
6 / 50 ⎟
⎟⎟
⎠
(17)
⎞
⎟
⎟
⎟
21 / 200 ⎟
⎟⎟
⎠
(18)
)
)
(19)
177
[(
C v,τ* = 1 + C v,Q
)
2 9 / 100
]
− 1 1/ 2
(20)
where Q is the mean volumetric discharge (m3/s) and Cv,Q is the coefficient of variation
of volumetric discharge. Notice that channel width has been removed from equations 18
and 20, and thus, these equations can be applied to channels where the site-specific
relationship between discharge and channel width is not available. However, this
approach only applies to channels that are effectively stable because the power-law
relationship between channel width and discharge with an exponent of 1/2 does not apply
to channels that are actively incising. Equations (17-20) can then be substituted into (1012) in order to quantify the long-term bedload sediment flux in an alluvial channel in
terms of the mean and the coefficient of variation of discharge. Although (17-20) do not
explicitly include seasonal variability in discharge, the coefficient of variation of
discharge is larger in regions with more seasonality in discharge, so the effects of
seasonality are implicitly included.
Channel slope was the remaining parameter in (17-18) required to convert the discharge
distribution to the Shields stress distribution, taking into account the channel geometry
and channel bed texture. Prescribing a channel slope independently of shear stress for
input into (18) can result in unrealistic Shields stress values at large discharges. Overestimating the Shields stresses during larger discharges would result in an overestimation
of long-term bedload sediment transport rates and effective discharge return periods.
Therefore, we calculated the channel slope such that the Shields stress for large
178
discharges was in an acceptable range. Using data from Church and Rood (1983), Marr et
al. (2000) showed that gravel- and sand-bed channels function at two distinct ranges of
Shields stress (i.e., 0.025 to 0.14 for gravel-bed channels and 0.5 to 2.0 for sand-bed
channels) during discharges of channel-forming significance. The return periods of the
channel-forming discharges included in the Church and Rood (1983) dataset range from
0.6 years to several years. Additional studies of channels located in humid to temperate
climates found that bankfull discharges, which can represent the channel-forming
discharge, have return periods on the order of one to several years with an average of 1.41.5 years (Leopold et al., 1964; Knighton, 1998). Based on these observations, we
calculated the channel slope such that the Shields stresses were within the range observed
by Marr et al. (2000) for discharges with return periods of one to several years. Equation
(14) was rearranged to solve for the channel slope that satisfies this condition:
⎛
⎛ ρ − ρf
⎜τ
b3/ 5 ⎜ s
⎜ * cf
⎜ ρf
⎝
S =⎜
3 / 10 3 / 5
⎜
Qcf n
⎜
⎜
⎝
⎞ ⎞
⎟d ⎟
⎟ ⎟
⎠ ⎟
⎟
⎟
⎟
⎠
10 / 7
(21)
where Qcf is the channel-forming discharge (m3/s) and τ*cf is the channel-forming
discharge Shields stress. A discharge with a specific return period can be calculated
directly from the lognormal distribution for discharge. However, choosing a unique value
for the prescribed channel-forming discharge return period is problematic because the
179
variation of this return period as a function of channel cross-section, channel bed texture
and hydrologic distribution is not well constrained. Instead of picking a single value for
the return period, we chose to test the sensitivity of our analytic solution to the prescribed
channel-forming discharge return period. Specifically, two channel slopes calculated
using the 1.5 and 6.0 year discharges were applied in our long-term bedload sediment
flux and effective discharge return period sensitivity studies.
2.2. Derivation of the diffusivity coefficient
The analytic equation for long-term bedload sediment flux can be applied to a number of
geomorphic problems. One application is to derive an analytic expression for the
diffusivity coefficient within the context of the diffusion model for the evolution of
alluvial channel profiles. A key assumption of diffusion modeling is that sediment flux is
linearly proportional to the local channel slope. Equations (10-12) show that long-term
bedload sediment flux is a nonlinear function of τ and therefore of S. As such, the
*
analytic expression for the long-term sediment transport rate must be linearized as a
function of slope in order to derive an expression for diffusivity. The diffusion model,
while it does not capture the full complexity of sedimentary basin evolution, is
nevertheless extremely useful and, when implemented in a model of sedimentary basin
evolution, permits the use of greater time steps compared to event-based models (e.g.
Paola et al., 1992; Ritchie et al., 1999; Coulthard et al., 2002).
180
Similarly to equations (10) and (11), the diffusivity κ that appears in (1) can, for gravelbed channels, also be written as an infinite series:
⎛ ρ − ρf
κ = ∑α ⎜ s
⎜ ρf
k =1
⎝
⎛ −⎛⎜ 3k −8 ⎞⎟
⎜ ⎝ 3 ⎠
⎞
⎟ gd d ⎜ τ *
⎟
⎜⎜ S 0
⎠
⎝
⎞
⎟
2
⎟ 1 + C v,τ*
⎟⎟
⎠
)
[1 − erf (β )]
(22)
⎛ ρ − ρf
κ = ∑α ⎜ s
⎜ ρf
k =1
⎝
⎛ −⎛⎜ 3k −8 ⎞⎟
⎜ ⎝ 3 ⎠
⎞
⎟ gd d ⎜ τ *
⎟
⎜⎜ S 0
⎠
⎝
⎞
⎟
2
⎟ 1 + C v,τ*
⎟⎟
⎠
)
[erf (β ) − erf (β )] + γ
(23)
∞
∞
⎛ ρ − ρf
γ= ⎜ s
⎜ ρf
⎝
(
(
3/ 2
1/ 6
⎞
⎛
⎟ gd d ⎜ 9.64[(1 + ε ) τ *cr ] (ετ *cr )
⎜
⎟
S0
⎝
⎠
⎛ ( 3 k −8 )( 3 k − 5 ) ⎞
⎜
⎟
18
⎝
⎠
⎛ ( 3 k −8 )( 3 k − 5 ) ⎞
⎜
⎟
18
⎝
⎠
2
⎡
⎤
⎞ ⎢ 1 1 ⎛⎜ ln[(1 + ε ) τ *cr ] − ln τ * + (1 / 2) ln(1 + C v,τ* ) ⎞⎟⎥
⎟ − erf ⎜
⎟⎥
2
⎟⎢ 2 2 ⎜
2 ln(1 + C v,τ* )1 / 4
⎟
⎠⎢
⎝
⎠⎦⎥
⎣
where g is the gravitational acceleration (m/s2), and α and β are the same as in (10)
(except that S is replaced by S0, the average slope of the basin that is being modeled with
the diffusion equation). Equations (22) and (23) are expressions for the diffusivity of
gravel-bed channels that include the mean and coefficient of variation of discharge, the
mean channel geometry, sediment texture, and the densities of sediment and water. The
diffusivity κ for sand-bed channels is given by:
⎛ ρ − ρf
κ = ⎜⎜ s
⎝ ρf
⎞ ⎛ 0.05d ⎞⎛ τ*5 / 2
⎟⎟ gd ⎜⎜
⎟⎟⎜
⎜
⎠ ⎝ 2C f ⎠⎝ S 0
(
⎞
⎟ 1 + C v,τ 2
*
⎟
⎠
)
15 / 8
(24)
181
The average slope S0 can be prescribed in the analysis or it can be estimated from
sediment and hydrologic parameters using (21).
2.3. Numerical and analytic solutions for the effective discharge
A third application for our analytic solution is to calculate the effective discharge and its
return period for gravel- and sand-bed channels. The peak of the effectiveness function is
the effective discharge that transports more sediment than all other discharge magnitudes.
For gravel-bed channels, setting the derivative of the product of (3) and (5) to zero and
solving for the Shields stress corresponding to the peak of the effectiveness function
results in an equation that cannot be solved analytically. Therefore, we chose to
numerically solve for the effective discharge for gravel-bed channels.
An analytic solution does exist for the effective discharge of sand-bed channels because
of the simpler form of the Engelund and Hanson (1967) transport equation. To solve for
the effective discharge we first substituted the analytic solution for the effective Shields
stress into (14). Equations (8) and (9) were used to replace the mean and variance of the
logarithms of Shields stress that are present in the analytic solution for the effective
Shields stress with the mean and variance of Shields stress for input into (14). We then
rearranged the resulting equation to obtain the following analytic solution for the
effective discharge for sand-bed channels:
182
⎛
⎛ρ −ρ
⎜ τ 1 + C 2 ( η −3 / 2 ) b 3 / 5 ⎜ s f
v, τ*
⎜ *
⎜ ρf
⎝
⎜
Qe =
3 / 5 7 / 10
⎜
n S
⎜
⎜
⎝
(
)
⎞ ⎞
⎟d ⎟
⎟ ⎟
⎠ ⎟
⎟
⎟
⎟
⎠
10 / 3
(25)
where Qe is the effective discharge (m3/s) and η is the exponent of the power-law
sediment transport function (i.e. 5/2 for the Engelund and Hansen (1967) transport
function). The resulting effective discharge for gravel- and sand-bed channels can then be
substituted into the following equation to solve for the return period of the effective
discharge:
TQe
(
⎛ ⎡
2
⎜ ⎢ 1 1 ⎛⎜ ln Qe − ln Q + (1 / 2 ) ln 1 + C v,Q
= ⎜1 − + erf ⎜
2
⎜ ⎢⎢ 2 2 ⎜⎝
2 ln 1 + C v,Q
⎣
⎝
(
)
)⎞⎟⎤⎥ ⎞⎟
−1
⎟⎟⎥ ⎟
⎠⎦⎥ ⎟⎠
(26)
where TQe is the effective discharge return period (units of the return period depend on
the sampling interval for Q).
3. Preliminary data analysis required for validation of model predictions
3.1. Climate effect on long-term bedload sediment flux
183
One of the goals of this study was to determine the geomorphic effectiveness of endmember climates. Given equations for the ability of a given channel to transport sediment
in a humid versus a more arid climate, we can start to place constraints on how channels
might change their geometry as climate changes. Statistical analysis of discharge data
from U.S. channels shows that with increasing aridity, the number of larger floods
relative to smaller floods increases (Turcotte and Greene, 1993). This observation has
since been supported by other studies (Molnar, 2001; Molnar et al., 2006). A comparison
of long-term bedload sediment flux for end-member climates must include this observed
covariation between the mean and variability of discharge as functions of climate.
Therefore, relationships between the variation in discharge, mean annual discharge, and
mean annual runoff were required to link the discharge distribution input into our analytic
solutions to climate.
Although previous studies have correlated the mean and the variation of discharge with
climate, further statistical analysis is required to obtain a relationship that can be applied
to our analytic solution. Molnar et al. (2006) fit power-law distributions to 155 stream
gauges within the United States and found a positive correlation between effective
precipitation and the exponent of a power-law distribution. In this study we assume that
discharge values are lognormally distributed. Power-law distributions are useful for
characterizing the tail of the frequency-size distribution of discharge but they are less
accurate at characterizing small, more common floods. McMahon et al. (2007) analyzed
more than 1000 channels globally and found that the relationship between mean annual
184
runoff (i.e. mean annual discharge divided by contributing drainage basin area) and the
coefficient of variation of annual discharge is approximately a power-law with an
exponent of –0.299. Averaging discharge over 365 days causes the coefficient of
variation for mean annual discharge to be less than the coefficient of variation for mean
daily discharge. A decrease in variation between daily and annual data results from
averaging out seasonal effects, large floods and days with zero discharge. However, daily
variations have significant effects on long-term bedload sediment transport and thus
should be included within the analytic solution. As such, we need to relate the coefficient
of variation of daily discharge to the mean annual discharge or runoff.
To do this, a regression analysis of the coefficient of variation of daily discharge (Cv,Qd)
versus mean annual runoff (R) was conducted for channel discharge data obtained from
the National Stream Water Quality Monitoring Network as part of the 1996 U.S.
Geological Survey Digital Data Series DDS-37 (Alexander et al., 1996). We analyzed
590 channel discharge records across the United States for mean daily discharge and
contributing drainage area. Only hydrograph records greater than 10 yr in duration were
included in the analysis, yielding a final number of 530 stations. Large drainage basins
are problematic because they are more likely to be impacted by nearby dams. The
presence of dams commonly decreases Cv,Qd relative to natural flow conditions by
reducing seasonally large flows and enhancing low flows (Magilligan and Nislow, 2005).
185
The coefficient of variation of daily discharge and the mean annual runoff were
calculated for each record and are plotted in Figure 1. When the data are plotted in loglog space there is a linear, inverse trend between the Cv,Qd and R, which implies a powerlaw relationship with a negative exponent. Although these data are strongly correlated,
there is significant scatter with many of the data points located near Cv,Qd and R values of
1 and a few data points with anomalously low Cv,Qd values. Channels with drainage basin
areas less than 5,000 km2 appear to plot toward greater Cv,Qd values than channels with
larger drainage basin areas for equivalent mean annual runoff values. One interpretation
of this pattern is that small drainages may not have a continuous base flow throughout the
year and as a result there is increased variability in discharge between storm and interstorm periods in small basins. We chose to only include drainage basins that were larger
than 5,000 km2 (shown as black asterisks) in our best-fit linear regression analysis to
reduce the influence of this basin-scale effect. The dashed line represents the resulting
least-squares fit to the logarithm of the data. Removing data points that represent
channels with smaller drainage areas improved the correlation coefficient from –0.45 to –
0.63. Visual comparison of the trend line with the overall data shows that Cv,Qd values
calculated for the arid end of the plot would be potentially underestimated by a factor of 2
to 3. This trend is explained by the fact that the majority of the data falls between R
values of 0.1 and 1 m, and thus, the trend line fits the data in the central region best (the
data located on the extreme ends of the R axis is not well fit). A least-squares regression
of binned data yields a fit that weighs each portion of the data throughout the range of
mean runoff values equally. The data were binned by averaging the points within each
186
bin spanning an order of magnitude variation of R. We then performed a least-squares
regression on the new data to calculate the solid line in Figure 1, i.e.
Cv,Qd = 0.868 R – 0.27
(27)
Equation (27), which is represented by the solid line in Figure 1, appears to more
accurately fit the data located on the arid end of the R axis than the regression for the
non-binned basins that are greater than 5,000 km2 (dashed line). However, it’s difficult to
assess if the new power-law improves the accuracy on the humid end of the R axis
because there are few large drainage basins in the U.S. that receive more than a meter of
runoff per year.
Now that we have a relationship between Cv,Qd and R, Q must be derived from R to
compare channels of similar hydraulic parameters using (17-20). Mean annual runoff is
simply the runoff per unit area of the drainage basin. As such, it can be multiplied by a
drainage area to yield Q . We chose a channel drainage area of 10,000 km2 because our
relationship between Cv,Qd and R was constrained by data collected from basins larger
than 5,000 km2 and it is an appropriate intermediate basin size. In our sensitivity studies,
mean annual runoff ranges between 0.001 and 1 m. This represents the range of mean
annual runoff that is well represented by the data in Figure 1.
187
4. Methods of Sensitivity Studies
4.1. Diffusivity sensitivity studies
We explored the sensitivity of the diffusivity to a prescribed range of values of Cv,τ* and
τ for both sand- and gravel-bed channels using (22-24). The advantage of performing
*
sensitivity studies in terms of the Shields stress distribution rather than the discharge
distribution is that a single Shields stress value can represent a range of channel
conditions. Shields stress values calculated from observed hydraulic parameters for
gravel- and sand-bed channels range between 0.025 to 0.14 and 0.5 to 2.0, respectively,
during discharges with channel-forming significance (Marr et al., 2000). In our sensitivity
study we chose mean Shield stress values that are less than or equal to the observed range
of Shields stress for channel-forming discharges because channel-forming discharges are
relatively large events. The range of Cv,τ* was directly calculated using (20) assuming that
Cv,Qd varies between 0.5 and 8.0 for U.S. channels. A grain size is required to convert
non-dimensional sediment flux to a sediment flux with units (e.g. m2/yr). We used grain
sizes of 0.02 and 0.001 m for gravel- and sand-bed channels respectively. The remaining
parameter for input into (22-24) was the average channel slope, prescribed to be 0.0035
for both gravel- and sand-bed channels. We chose this value because it is a middle ranged
depositional basin slope that is lower than typical alluvial fan slopes but on the high end
for coarse grained river gradients. While the choice of the basin average slope does affect
188
the magnitude of the diffusivity, it does not significantly affect the first order behavior of
the diffusivity as a function of mean Shields stress.
Thus far we have assumed that the diffusivity was only affected by changes in the
magnitude and frequency of discharges. However, limiting shear stresses to near
threshold conditions during large discharges (i.e., the self-forming channel of Parker
(1978)) can significantly affect the relationship between diffusivity and the mean Shields
stress in gravel-bed channels. In order to test the effect of limiting shear stress to account
for channel bank widening on diffusivity we compare the results for calculating the
diffusivity using (22) and (23). Limiting the shear stress should cause the diffusivity to
decrease below the diffusivity of an equivalent channel with sufficient bank cohesion to
sustain shear stresses that are well above critical.
Following the description of the results of our sensitivity studies, we compare published
estimates of diffusivities for alluvial channels to those calculated with our analytic
solution. Recently, Moshe et al. (2008) calculated diffusivities for several alluvial
channels draining into the Dead Sea in Israel that have been adjusting to falling base level
over the past 150 years. We used channel morphology and hydraulic parameters reported
by Moshe et al. (2008) and Bowman et al. (2007) for input into our model (Table 1). For
the parameters that were not explicitly given, we used parameter values that fell within
the range of typical observed values for the region (labeled with a c in Table 1).
189
4.2. Long-term bedload sediment transport sensitivity studies
Sensitivity studies for the long-term bedload sediment flux as a function of mean annual
runoff and the coefficient of variation of daily discharge were conducted for both graveland sand-bed channels. Equations (18) and (20) were used to convert the mean and
variation of discharge into the mean and variation of Shields stress using the hydraulic
parameters given in Table 2. We chose a middle-ranged b value of 5.0 s1/2/m1/2 to relate
channel width to discharge. The final parameter required for input into (18) was channel
slope. Fixing channel slope for all mean annual runoff is a simplification because the
effective discharge is a function of the mean and variability of discharge. However, our
goal was to compare alluvial channels of a single prescribed texture and channel slope
(i.e. with values chosen to yield realistic hydrologic scenarios) across a range of mean
annual runoff conditions instead of attempting to keep the Shields stress distribution the
same as mean annual runoff varies.
The fixed channel slope calculated using (21) is the slope at which the channel-forming
discharge (Qcf) generates the prescribed Shields stress (τ*cf). As such, the Shields stress
distribution of our channels, which is a function of channel slope, is sensitive to these two
parameters. The channel-forming discharge was calculated from the lognormal
distribution using a prescribed return period (i.e., either 1.5 or 6.0 years) and a prescribed
mean annual runoff. Channel-forming discharge return periods reported in the Marr et al.
190
(2000) study and others are predominantly for channels located in humid to temperate
climates. Therefore, we chose the hydrologic distribution used to calculate the channelforming discharge to have a prescribed mean annual runoff of 0.5 m. The coefficient of
variation of daily discharge for the hydrologic distribution was then calculated using (27).
Equation (21) also requires a prescribed Shields stress (τ*cf) for the channel-forming
discharge. Shields stress values range between 0.025 to 0.14 and 0.5 to 2.0, for graveland sand-bed channels respectively, during discharges with channel-forming significance
(Marr et al., 2000). We chose mid-ranged channel-forming discharge Shield stresses of
0.07 for gravel-bed channels and 1.0 for sand-bed channels. These shear stress values are
appropriate for calibrating the channel slope because shear stress distributions are
realistic for all considered R-Cv,Qd pairs characteristic of U.S. channels (i.e., the channelforming Shields stresses calculated for channels located in end-member climates fall
within the observed range of Shields stresses). However, channel-forming Shields
stresses picked near extreme ends of the observed Shields stress ranges do not yield
realistic Shields stress distributions for end-member climates.
In addition to climate change, another process that affects the long-term bedload
sediment fluxes is channel width adjustment to large discharges (i.e., the self-forming
channel of Parker (1978)). Limiting the shear stresses produced by large discharges
should reduce the long-term sediment fluxes if the contribution of large discharges to the
191
total sediment flux is significant. As such, we test the sensitivity of long-term sediment
fluxes to regulating shear stress by comparing the results for (10) and (11).
Following the description of the sensitivity analysis, we compare our analytic solution for
long-term bedload sediment fluxes to published data. Validating model predictions
against observed data is essential to any quantitative analysis. Few datasets exist,
however, during which continuous bedload transport rates have been sampled for all
channel flow events over time scales of a few years to decades. Studies do exist where
bedload transport rates were measured briefly during flows of varying magnitude over
several years in order to develop power-law relationships between bedload sediment
transport rates and discharge (i.e. rating curves). One of the most complete studies to date
was conducted for 33 gravel-bed channels in Idaho (King et al., 2004). The parameters
applied to this analysis are given in Tables 1, 7 and 8 of King et al. (2004). Discharge
time series data were obtained from the National Water Information System online
hydrological database (NWISWeb, http://waterdata.usgs.gov/nwis/rt) maintained by the
United States Geological Survey and from the Rocky Mountain Research Station website
(http://www.fs.fed.us/rm/boise/research/watershed/BAT/index.shtml) maintained by the
USDA Forest Service. Observed long-term bedload sediment transport rates were
calculated by numerically integrating the product of the rating curves given by King et al.
(2004) and the discharge time series and then dividing by the number of days of data.
King et al. (2004) published channel cross-section data that we used to compute channel
width. Equations (17) and (19) were used to calculate the mean and variation of Shields
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stress. We then compared the long-term bedload sediment transport rates calculated from
the results of King et al. (2004) to the rates predicted by our analytic solution.
4.3. Effective discharge return period sensitivity studies
The relationship between the effective discharge and climate is not well constrained
because few bedload studies exist for semiarid to arid drainages. Therefore, we also
tested the sensitivity of the effective discharge return period to the mean annual runoff
and the coefficient of variation of discharge. We applied the hydraulic parameters
reported in Table 2 and a b value of 5.0 s1/2/m1/2 (i.e., to relate channel width to
discharge) as input to these equations. As in the sensitivity study for long-term bedload
sediment flux, fixed channel slopes were calculated such that the 1.5 and 6.0 year
discharges yield realistic Shields stresses.
The effective discharge, and by extension the effective discharge return period, is
sensitive to the relationship between discharge and sediment flux. Limiting the shear
stress of larger-magnitude lower-frequency discharges should cause the effective
discharge return period for gravel-bed channels to decrease. Therefore, we also tested the
sensitivity of the effective discharge return period to limiting the shear stress.
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Following the results of the sensitivity studies in the next section we compare effective
discharge and effective discharge return periods of gravel-bed channels reported by Barry
et al. (2008) and Torizzo and Pitlick (2004) to effective discharges and effective
discharge return periods predicted by our effectiveness function. The parameters used for
this study were found in Tables 1, 7 and 8 of King et al. (2004) and Table 1 of Torizzo
and Pitlick (2004). Discharge hydrographs were obtained from the same website
locations as reported in the long-term bedload sediment flux sensitivity studies.
5. Results of sensitivity studies
5.1. Diffusivity sensitivity studies
Adding a threshold term in the transport function appears to have the greatest effect on
how diffusivity varies with mean Shields stress when the results for sand- and gravel-bed
channels are compared (Figure 2). In the case of the sand-bed channels (dashed lines),
diffusivity values increase by approximately two orders of magnitude as Shields stress
values increase by one order of magnitude from 0.2 to 2. Diffusivities for gravel-bed
channels (solid lines) vary over a wider range, i.e. several orders of magnitude as mean
Shields stress increases by approximately the same amount from 0.01 to 1. The effect of a
threshold term appears to be less significant as mean Shields stress increases above 0.06
and the relationship between diffusivity and mean Shields stress returns to a power-law
behavior.
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Although diffusivity values are most strongly correlated with mean Shields stress, the
coefficient of variation of Shields stress also exerts a significant effect on diffusivity for
both types of channels values (Figure 2). The increase in diffusivity values is more
sensitive to Cv,τ* for gravel-bed channels when mean Shields stress is less than or equal to
the critical Shields stress. A gravel-bed channel with a mean Shields stress well below the
critical Shields stress but a high Cv,τ* value can transport significantly more sediment than
one with a low Cv,τ*. Sand-bed channels appear to consistently influenced by the
coefficient of variation of Shields stress across the range of mean Shields stress. A
comparison between gravel- and sand-bed channels at the high end of mean Shields stress
values reveals that sand-bed channels are more sensitive to Cv,τ* values compared to
gravel-bed channels. This occurs because the exponent of the coefficient of variation
term in (24) is larger than the equivalent exponents in (22-23).
A third process that was considered for its effects on the diffusivity for gravel-bed
channels was the channel width adjustment when Shields stresses exceed a critical
threshold value (Figure 2). A comparison of self-forming channels (thick, solid lines) to
channels able to sustain Shields stress values significantly greater than critical (thin, solid
lines) reveals that diffusivity is unaffected when both mean Shields stress and the
coefficient of variation of Shields stress are low (e.g. Cv,τ* of 0.2). At higher values of
both the mean and coefficient of variation of Shields stress, self-forming channels have
consistently lower diffusivities. The diffusivity of self-forming channels appears to
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converge to the instantaneous transport rate for (1+ε) τ*cr at the high end of mean Shields
stress and is essentially independent of mean Shields stress. The rate that diffusivity
converges appears to be dependent on the coefficient of variation of Shields stress.
The above sensitivity study results show a broad range of diffusivity values that depend
primarily on mean Shields stress. The diffusivity calculated with (22) for a gravel-bed,
single-thread channel that drains a 1,000 km2 basin and that has a mean annual runoff of
100 mm is approximately 5.3x101 m2/yr when the average basin slope is on the order of
10-3 (Table 3). A gravel-dominated channel that drains a larger basin (10,000 km2)
located in wetter climate that has a mean annual runoff of 1.0 m yields in a diffusivity of
approximately 2.67x106 m2/yr. The 5 orders of magnitude increase in diffusivity between
the gravel-bed channels is mainly caused by the existence of a threshold slope condition
for transport. For sand-dominated channels characterized similar drainage areas, a
channel slope of 10-4, channel bed sediment comprised of quartz sand with a dominant
grain size of 1 mm, and mean annual runoff values that range between 0.1 and 1.0 m, the
resulting diffusivities range between 2.11x104 and 7.13x105 m2/yr (i.e., 0.021 and 0.71
km2/yr).
We compared diffusivities calculated using (22) with diffusivities reported in the
literature to assess the validity of our analytic solution. Calculated diffusivities for
smaller-sized basins were within a factor of 4 of the diffusivity values reported by Moshe
et al. (2008) (gray rows within Table 1). Diffusivities calculated for larger basins are as
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much as 2 orders of magnitude greater than the reported values. For each basin
considered in our study, we divided the rainfall index or annual volume of rain delivered
to a basin reported by Moshe et al. 2008 by the contributing area to calculate the effective
mean annual runoff. Ideally, the mean annual runoff should be directly calculated from a
hydrograph record. However, few basins in this region have been monitored with stream
gauges. Over-estimation of diffusivities for large basins may result from substituting
rainfall as a proxy for runoff because larger basins may have a significant portion of their
rainfall lost to infiltration and evaporation. In order to correct for the basin scale effect,
we used a power-law relationship with exponents of −0.3 to −0.5 to calculate the mean
annual runoff as a function of drainage area for semi-arid to arid basins (Farquharson et
al., 1992; Baker, 2006). The power-law coefficient was determined from the effective
mean annual runoff calculated for the smallest basin in the Moshe et al. (2008) data.
When a power-law with an exponent of −0.3 is used to predict mean annual runoff, the
calculated diffusivities show similar misfit to the diffusivities calculated using the rainfall
index. However, when a power-law exponent of −0.5 is used to predict the mean annual
runoff, the calculated diffusivities for larger basins show improved fit to the reported
diffusivities.
5.2. Climate effects on long-term sediment transport
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Figure 3 plots the results of solving (10-12) over a range of mean annual runoff values
with a fixed channel slope and texture for both gravel- (Figure 3A,C) and sand- (Figure
3B,D) bed channels. The gray curves represent constant Cv,Qd values of 0.5, 2 and 8,
respectively. Both gravel- and sand-bed channels (solid gray lines) exhibit a positive
correlation between long-term bedload sediment flux and mean annual runoff. Results for
a gravel-bed channels show that the relationship between long-term bedload sediment
flux and mean annual runoff is not a simple power-law relationship, which is evident
from the general form of (10) (Figure 3A). Increasing Cv,Qd leads to an increase in longterm sediment flux. Bedload sediment flux is effectively zero for a Cv,Qd of 0.5 at low
mean annual runoff values below 0.1 m. If the curves for Cv,Qd of 8 and 2 are projected
beyond a mean annual runoff of 1m, a point is reached where increasing Cv,Qd decreases
long-term bedload sediment fluxes. The location of this cross-over point is sensitive to
the threshold of entrainment and the Shields stress distribution. Increasing the mean
Shields stress by increasing the channel slope or decreasing the grain size would shift the
cross-over point to a lower mean annual runoff. The relationship between Cv,Qd and mean
annual runoff is less complicated for sand-bed channels (Figure 3B). The general form of
(12) is a power-law that lacks a threshold term, and therefore, the relationship between
sediment flux and mean annual runoff is a straight line in log-log space. Increasing Cv,Qd
leads to a decrease in long-term bedload sediment flux over the entire range of mean
annual runoff because all discharge magnitudes produced in the sand-bed channel are
able to entrain sand. Increasing the Cv,Qd increases the frequencies of the larger
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discharges, but this increased contribution is not enough to offset the loss of work
performed by mid-range discharges as the mode of the discharge distribution decreases.
Increasing channel width in response to discharges that exceed the threshold for
entrainment (i.e. the channels are self-forming) also has an effect on the relationship
between long-term bedload sediment flux and mean annual runoff in gravel-bed channels
(dashed-gray lines in Figure 3A). A comparison of self-forming channels to channels that
are able to sustain Shields stresses well above the threshold for entrainment shows that
limiting the shear stress of larger floods decreases the long-term bedload sediment flux.
The divergence in long-term bedload sediment transport rates between the two types of
channels appears to be negligible at low mean annual runoff values when Cv,Qd is less
than 8. However, the divergence is significant as mean annual runoff increases toward
1m. If mean annual runoff continued to increase above 1m the long-term bedload
sediment flux would approach the instantaneous sediment flux for the critical Shields
stress. The mean annual runoff value at which the long-term rate approaches a limit is
dependent on the mean Shields stress. Limiting Shields stress to near critical values
during large discharges also shifts the location where increasing Cv,Qd begins to decrease
the long-term bedload sediment flux values toward a lower mean annual runoff value.
This occurs because the contribution of larger discharges is reduced by limiting the
Shields stress to near the critical Shields stress.
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For channels located in the United States, Cv,Qd is negatively correlated with mean annual
runoff. The dot-dashed black lines in Figure 3 represent the long-term bedload sediment
fluxes for U.S. channels with similar channel parameters to those chosen for our study.
Long-term bedload sediment flux values increase with mean annual runoff for both
channel types because increasing the mean Shields stress has a greater effect on the longterm sediment flux than decreasing the variation in Shields stress. Overall, the negative
correlation between Cv,Qd and mean annual runoff causes the change in long-term
sediment fluxes with decreasing mean annual runoff for gravel-bed channels to be less
significant than the change predicted for a channel that maintained a constant low value
(i.e., ≤ 2) of Cv,Qd. However, negatively correlating mean annual runoff and Cv,Qd does
lead to an increase in the magnitude of change in sediment flux for sand-bed channels.
We also tested the sensitivity of long-term bedload sediment flux to calibrating the
channel slope based on a channel-forming discharge return period. Channel slopes were
calculated such that the Shields stress of a discharge of a given return period fell within
the observed range for channel-forming discharges. The higher the discharge return
period, the lower the channel slope must be to yield the prescribed Shields stress. Figures
3A,B represent the results for a channel-forming discharge return period of 1.5 years,
while Figures s 3C,D represent the results for the 6.0 year channel-forming discharge.
Increasing the return period for the channel-forming discharge by a factor of 4 lead to
less than a factor of 2 decrease in long-term sediment flux for curves that represent
gravel-bed channels with high Cv,Qd values. For lower Cv,Qd values located toward the
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humid end of the mean annual runoff axis long-term sediment flux decreased by a factor
of 2 or more. Sand-bed channels showed less than a factor of 2 decrease in long-term
sediment flux regardless of the Cv,Qd value. Although the magnitude of long-term
sediment transport rates was affected by the choice of channel-forming discharge, the
first-order behavior did not change.
The curves in Figure 3 are profiles parallel to the mean annual runoff axis from the
continuous parameter space that is shown in Figure 4 for gravel- (A) and sand- (B) bed
channels. The advantage of the plots in Figure 4 is that effect of changes to the exponent
for the power-law relationship between mean annual runoff and Cv,Qd or a uniform
change in Cv,Qd can be more easily visualized in a 2-dimensional space. The power-law
relationship between mean annual runoff and Cv,Qd for U.S. channels is represented by the
solid white line. A decrease in the power-law exponent or an increase in the negative
correlation is equivalent to an increase in slope of the solid white lines in Figure 4. A
uniform change in Cv,Qd across climates of varying aridity is a vertical axis shift in the
trend line. To first order, long-term bedload sediment fluxes appear to be more affected
by variations in mean annual runoff than by variations in Cv,Qd. Upon inspection of
Figure 4, neither a rotation nor a uniform shift in the covariation between mean annual
runoff and Cv,Qd for the U.S. will allow channels located in arid end-member climates to
transport more sediment over a period of time than equivalent channels in humid endmember climates. A climatically or anthropogenically forced increase in Cv,Qd for a
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gravel-bed channel could lead to an increase in long-term bedload sediment transport
rates if the mean runoff value was unaffected. Increasing the Cv,Qd in sand-bed channels
will lead to a decrease in long-term bedload sediment fluxes
Following the sensitivity studies we also compare our analytic solution for long-term
bedload sediment fluxes to values calculated from field data collected by Barry et al.
(2004) from gravel-bed channels (Figure 5). Long-term sediment fluxes predicted by our
analytic solution (asterisks) are on average comparable to field-measured data but also
show scatter about a one-to-one line. One explanation for the scatter is that general
transport equations are less accurate at predicting instantaneous sediment fluxes given
mean channel conditions than site-calibrated sediment rating curves. We also plotted the
long-term bedload sediment transport rates predicted by integrating the product of the
Barry et al. (2004) sediment transport equation and a log normal distribution (triangles)
against the observed rates. Predicted rates still show scatter, but the variation about the
one-to-one line is significantly less, and thus, the accuracy of predicted long-term
bedload sediment transport rates appears to be primarily affected by the choice of
sediment transport equation. Another issue to consider is the spread of the observed
bedload sediment transport rates. Although the observed bedload sediment transport rates
cover 2-orders of magnitude, the majority of the data represent small drainages (< 5,000
km2). Also, the data collected by Barry et al. (2004) comes from a sufficiently small
study area that mean annual runoff and Cv,Qd are approximately uniform over the study
area. We feel that the correlation between our analytic solution predictions and observed
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bedload sediment transport rates would improve with more data from other climatic
regions and basin sizes that cover a wider range of long-term bedload sediment transport
magnitudes.
5.3. Climate effects on effective discharge return period
A third application of our analytic solution was to calculate the return period of the
effective discharge. Geomorphic effectiveness can be quantified using a variety of
measures, but in alluvial channels bedload sediment flux is the most appropriate measure
to adopt. The return period of the effective discharge provides an approximate time scale
for geomorphic “equilibrium” because, if the return period is long (e.g. decades to
centuries), the landscape can be considered to be “recovering” from the impacts of
discharges of that return period in the intervals of time between events of that size.
Sensitivity studies were conducted for gravel- and sand-bed channels using the same
parameters listed in Table 2 in order to determine the return period of the effective
discharge. The slope of the discharge effectiveness function is the product of the
probability density function for discharge and transport equation as a function of
discharge. The discharge beneath the peak of the discharge effectiveness function
represents the effective discharge that is responsible for transporting the most sediment
over a given interval of time (Wolman and Miller, 1960). In the case of gravel-bed
channels, for simplicity we chose to numerically solve for the peak of the discharge
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effectiveness function for each mean discharge and Cv,Qd pair. For sand-bed channels, we
used (26) to solve for the effective discharge.
The return period for a discharge that is greater or equal to the effective discharge is
negatively correlated to the coefficient of variation of daily discharge and mean annual
runoff for gravel-bed channels (Figure 6A). The effective discharge return period
decreases by several orders of magnitude across the range of mean annual runoff. The
decrease in the effective discharge return period over the range of Cv,Qd depends on the
mean annual runoff. As Cv,Qd increases for channels with low mean annual runoff, the
frequency of the larger discharges increases, and thus, the return period of discharges that
are greater than or equal to the effective discharge decreases. At higher mean annual
runoff, the lower-magnitude discharges near the mode of the distribution are able to
exceed the threshold for transport. Therefore, increasing the frequency of larger
discharges near the tail of the distribution does not significantly decrease the return
period for the effective discharge. Conversely, at low mean annual runoff the discharges
close to the mode of the distribution might not exceed the threshold for entrainment, and
thus, increasing the frequency of larger storms leads to a decrease in the return period of
the effective discharge.
Gravel-bed channels that are considered self-forming maintain a constant Shields stress
for discharges that exceed the threshold for entrainment. Comparing Figures 6A and 6B
shows that the return period for the effective discharge decreased in the region where
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both mean annual runoff and the coefficient of variation of daily discharge are high when
channels are self-forming. A lesser decrease also occurred in the region of high
coefficient of variation and lower mean annual runoff. The region of low coefficient of
variation beneath the solid white line appears to be unaffected by limiting Shields stress.
Overall, decreases in the return period for the effective discharge were less than a factor
of 5. The small degree of change between Figures 6A and 6B suggests that the Shields
stress of the effective discharge is generally less than the (1+ε)τ*cr for gravel-bed
channels with lower mean annual runoff and coefficient of variation of daily discharge.
Excluding a threshold term in the sediment transport equation for sand-bed channels has
a significant effect on how the effective discharge return period varies with the mean and
the variation of discharge. The effective discharge return periods for sand-bed channels
show a positive correlation with Cv,Qd and a weaker correlation with mean annual runoff
when mean annual runoff is high (Figure 6C). Increasing the frequency of larger
discharges causes the effective discharge return period to increase. Nash (1994) derived
solutions for the effective discharge as a function of the mean and standard deviation of
discharge for lognormal and normal distributions. Our results are in agreement with those
of Nash (1994) that the return period of the effective discharge has a stronger dependency
on the variation than the mean of discharge when mean annual runoff is high.
At lower mean annual runoff values the effective-discharge return period for sand-bed
channels switches from being primarily dependent on Cv,Qd to being primarily dependent
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on mean annual runoff. The lognormal distribution does not assign any probability to
zero discharge, which causes the effective discharge return period to be too low for
channels with low mean annual runoff. Ephemeral channels in the semi-arid to arid
southwestern U.S. can spend more than half a year without flow, and thus, the effective
discharge return period should be greater than that interval. Hydrographs for ephemeral
U.S. channels that represent the arid end of mean annual runoff values were analyzed to
determine the relationship between the maximum period of consecutive days without
flow and mean annual runoff. If the computed return period was less than the observed
period for the given mean annual runoff, then the computed return period was set to the
observed maximum period of dry days. We found that U.S. channels with arid to semiarid drainage basins that were greater than or equal to 10,000 km2 still have discharge
running through their channels a few days a year. As such, the return period of the
effective discharge for sand-bed channels near the arid end of the mean annual runoff
axis is on the order of 0.1 years.
Although a broad range of effective-discharge return periods occurs as the mean annual
runoff and Cv,Qd are varied, channels tend to occupy a discrete region that is represented
by the solid white line in Figure s 6A, 6B and 6C. Effective discharge return periods for
the sand-bed channel case are less than one year over the entire range of mean annual
runoff. The threshold for entrainment of sand was set to zero by the Engelund and
Hansen (1967) transport equation, and thus, the return period for the effective discharge
is low compared to an equivalent gravel-bed channel. Effective discharge return periods
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for the gravel channels were calculated to be between less than a year to hundreds of
years, with effective discharge return periods positively correlated with aridity.
The return periods predicted for the arid end of the axis may be bias toward higher values
because the channel slope was prescribed such that the 1.5-year discharge for a channel
with a mean annual runoff of 0.5 m had a particular Shields stress. Channels located in
basins within more arid climates do not experience flows of channel-forming significance
with the same frequency as channels in temperate climates. Figure 6D shows the effect of
increasing the assumed channel-forming discharge return period from 1.5 to 6.0 years for
both gravel- and sand-bed channels. For gravel-bed channels, increasing the assumed
channel-forming discharge return period by a factor of 4 led to a comparable increase in
the effective-discharge return period. The increase in the effective-discharge return
period appears to be dependent on mean annual runoff. Increasing the assumed channelforming discharge return period did not significantly affect the effective-discharge return
periods for sand-bed channels.
We also tested how accurately our solution predicted both the effective discharge and
effective-discharge return period for Idaho and Colorado gravel-bed channels compared
to the data reported Torizzo and Pitlick (2004) and by Barry et al. (2008). Most of the
predicted effective discharge points are located within a factor of four of the one-to-one
line and appear to be systematically higher than the observed effective discharges (Figure
7A). The scatter in points about the one-to-one line might result from the limited
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accuracy of a general transport equation for predicting sediment fluxes based on mean
channel parameters. The predicted effective-discharge return periods show a wider spread
about the one-to-one line with more of the points located above the one-to-one line
(Figure 7B). The systematic over-prediction of the effective discharge return period arises
from overestimating the effective discharge. However, some of the predicted return
periods are within a factor of 2 of the observed return periods. These results may improve
if the effective discharge and effective discharge return period are calculated by binning
grain size instead of using a single representative grain size. The d50 may be coarser than
the grain size that contributes the most toward the total bedload sediment transport rate
during the effective discharge, and thus, greater discharges are required to transport
sediment.
6. Discussion
6.1. Diffusivity
Over the range of observed Shields stress values for sand- and gravel-bed channels,
diffusivities are up to two orders of magnitude higher and lower than the range of
published diffusivity values (i.e., between 104 to 105 m2/yr) reported by other studies
(Begin et al., 1981; Flemings and Jordan, 1989; Paola et al., 1992; Marr et al., 2000).
Direct comparison between our results and Marr et al. (2000) and Paola et al. (1992) is
difficult because their diffusivities where derived independently of specified grain size
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and channel slope. Diffusivities are within the range of reported values when mean
Shields stress and the coefficient of variation of daily Shields stress are between 0.020.03 and 0.2-0.4 for gravel-bed channels. Diffusivities for sand-bed channels are on the
order of 105 m2/yr when mean Shield stress is less than 0.6. While our results support the
previously reported diffusivity values, diffusivity values can vary over a range greater
than an order of magnitude depending on the channel slope, grain size and hydrologic
distribution. Therefore, it is necessary to use these parameters to calculate the diffusivity.
Using the analytic equations for diffusivity given by (22-24), landscape evolution models
can now quantify the effects of both the mean annual runoff and the coefficient of
variation for discharge within transport-limited channels. Commonly, landscape
evolution models and diffusion models utilize a representative flood discharge (Tucker
and Slingerland; 1996; Marr et al., 2000). This representative discharge is not well
defined and does not account for hydroclimatic variability. By incorporating both the
mean and the variability of discharge, the full discharge distribution can be applied to
calculating the long-term sediment flux without having to numerically resolve individual
flood events, which can be computationally prohibitive for some models when run over
geologic time scales.
6.2. Climate effects on sediment transport and the effective discharge return period
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Given a channel segment that has a constant slope and grain size distribution as climatic
conditions shift from wetter to dryer periods (the latter characterized by flashier
discharges), the long-term bedload sediment flux decreases for both gravel- and sand-bed
channels (represented by the solid white line of Figure 4). If the variation in discharge is
held constant, then decreasing the mean annual runoff within a drainage basin should
decrease the frequency of greater-than-threshold discharge events that contribute to the
long-term sediment flux of the basin, especially for gravel-bed channels. This conclusion
is consistent with those of Molnar et al. (2006). Increasing the discharge variability
across the range of considered mean annual runoff values for gravel-bed channels is
predicted to increase long-term bedload sediment transport rates and partially offset the
effects of decreasing the mean annual runoff. Across the entire range of mean-annual
runoff in sand-bed channels and at very high mean annual runoff (i.e., greater than 1m) in
gravel-bed channels, long-term bedload sediment fluxes decrease with increasing
discharge variability (Figure 4). For the lognormal distribution, increasing the variability
while maintaining a constant mean value causes the mode to shift toward smaller
discharges. Long-term bedload sediment transport rates decrease in such cases because
the increase in frequency of lower rates is not offset by the increase in frequency of
higher rates.
Figure 4 shows that a gravel-bed channel located at the arid end of the mean-annualrunoff axis could achieve a long-term bedload sediment flux that is on the order of an
equivalent channel with a temperate mean annual runoff if the Cv,Qd is sufficiently high.
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However, mean annual runoff and Cv,Qd have been demonstrated to covary, which places
a limit on the increase in Cv,Qd with increasing aridity (McMahon et al., 2007). As a
result, the increase in the frequency of larger discharges is not sufficient to offset the
component of sediment transported by lost by shifting the mode of the discharge
distribution to lower discharges with decreasing mean annual runoff. As such, the
threshold for entrainment would have to be much larger than the threshold considered in
this study to allow an increase in long-term bedload sediment transport rates with
increasing aridity.
Conversely, the return period for the geomorphically-effective discharge is inversely
proportional to mean annual runoff. Sand-bed channels have effective flow return periods
on the order of a year or less over the range of considered mean annual runoff. Gravelbed channels in arid to hyper arid climate end-members with mean annual runoff values
of less than 10 mm have effective discharge return periods on the order of hundreds of
years. As the mean annual runoff of a region increases, the return period of the effective
discharge decreases because the mode of the distribution moves toward larger discharges
that are capable of entraining sediment. Humid climate end-members with mean annual
runoff values of greater than 1000 mm have effective discharge return periods one the
order of 0.1 years (i.e., scale of a few weeks to a month). These results are in agreement
with the return periods calculated by Torizzo and Pitlick (2004) for gravel-bed channels
with temperate mean annual runoff (i.e., 0.4-0.5 m) and D50 that range from 23 to 80 mm.
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They found that the effective discharge was exceeded at least 1.5 to 11.3 days out of the
year.
In addition to the mean and variation of the discharge distribution, channel width
response to large discharges within gravel-bed channels can affect long-term bedload
sediment transport rates. At high mean annual runoff values, long-term bedload sediment
transport rates converge toward the instantaneous sediment transport rate calculated for a
Shields stress of (1+ε)τ*cr. This behavior suggests that self-forming channels may have a
natural upper limit to increasing long-term sediment transport rates with increasing mean
annual runoff as long as the channel remains stable (i.e. does not undergo active
entrenchment). However, the mean annual runoff would have to be greater than 1m for
the channels considered in our sensitivity study to reach the long-term sediment flux
limit.
Self-regulation of Shields stress values via channel width adjustment within gravel-bed
channels appears to have less of an effect on the effective discharge return period than the
long-term sediment flux. One explanation is that the effective discharge generally yields
a Shields stress that is less than or equal to 1.2 to 1.4 times the critical Shields stress. As
such, placing an upper limit on Shields stress did not significantly shift the effectiveness
peak. However, increasing the exponent in the sediment transport equation may lead to
greater effective discharges and effective discharge return periods. If the effective
discharges sufficiently increased to yield Shields stresses beyond 1.4 times the critical
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Shields stress, then placing an upper limit on Shields stress would have a more significant
effect on the effective discharge return period.
It is important to emphasize that our results assume that the sediment flux always follows
the Wiberg and Smith (1989) and Engelund and Hansen (1967) relationships. However,
gravel-bed channels that develop significant channel armoring might violate this
assumption. Channels with a mixture of bed grain-sizes and low or decreasing sediment
supply can develop a strong vertical sorting in channel bed grain size under constant
discharge conditions or during the falling limb of a storm hydrograph if the period of
decreasing discharge is not brief (Hassan et al., 2006). Covering finer subsurface
material with a layer of coarse material prevents sediment transport during more frequent,
lower magnitude discharges, and thus, the channel is considered detachment-limited
instead of transport-limited during flows that might be sufficient to entrain grain sizes
less than or equal to the D50 grain size. A second effect of channel armoring is to increase
the power-law exponent that relates sediment flux to discharge. Finer sediment in the
channel bed subsurface is exhumed and mobilized in addition to the coarse surface
material during large discharges capable of eroding the channel armor. Emmett and
Wolman (2001) observed that channel armoring can lead to rating curve exponents up to
5, which is significantly greater than the 3/2 exponent for excess Shields stress (i.e., 9/10
exponent for discharge) assumed by our chosen sediment transport equation. Increasing
the power-law exponent for the relationship between sediment flux and Shields stress
213
causes the long-term bedload sediment transport rates and the effective discharge return
period to increase.
Our results for the long-term bedload sediment flux and effective discharge return period
are also dependent on the type of distribution that is used for discharge. We applied a
lognormal distribution that accurately approximates the frequency of smaller discharges
that are close to mean values for channels in North America. Other distributions (e.g.
power-law distribution) have been found to more accurately fit the distal tail of the
discharge distribution that represents very large, infrequent discharge magnitudes
(Turcotte and Greene, 1993; Molnar et al., 2006). However, the lognormal distribution is
the best distribution available to accurately fit all parts of the distribution. Future research
might involve a hybrid of the two distributions to obtain the best representation of the
discharge distribution.
Our results for the return period of the effective discharge have implications for relating
basin-averaged erosion rates of different timescales. Kirchner et al. (2001) found that
basin-averaged erosion rates measured from gravel-bed channels in Idaho representative
of decadal timescales based on the total sediment load are an order of magnitude lower
than basin-averaged erosion rates that represent timescales of thousands to millions of
years. The decrease in long-term erosion rates was interpreted as an effect of sampling
over a short enough timescale such that decadal-scale erosion rates did not capture a
large, infrequent storm event that accounts for 90 percent of the millennial timescale
214
geomorphic work. Our results for modern climate conditions suggest that the most
effective geomorphic event in gravel-bed alluvial channels located in semi-arid to humid
climates occurs on a time scale that is much shorter than a millennium. Sand-bed
channels appear to experience effective discharges on timescales of on the order of one
year. Unless alluvial channel cross-sections become perturbed by a phase of
entrenchment (i.e., channel disequilibrium) or the lognormal distribution greatly
underestimates the frequency of large discharges at the tail of the distribution, bedload
sediment transport rates measured over a period of a few years to a decade should be
representative of the bedload contribution toward the long-term basin-averaged erosion
rates for drainage basins located in temperate to humid climates, provided that the
vegetation of the drainage basin stays constant. Erosion rates can increase dramatically
following fires and other perturbations that lead to changes in erosivity and it may be that
some of the episodicity invoked by Kirchner et al. (2001) to explain their data is related
to changes in vegetation and soils, including post-fire effects. However, our results
suggest that hydrologic variability alone cannot result in effective discharge return
periods on the order of a millennium or more.
7. Conclusions
In this study, we have derived analytic solutions for diffusivity, long-term bedload
sediment flux and the effective discharge return period. Our results showed that the long-
215
term bedload sediment flux of alluvial channels is highly dependent on the hydroclimatic
regime. As such, diffusivities should be a function of both the mean and variation of the
discharge distribution. We also showed that the mean and variation of discharge are
negatively correlated for U.S. channels. Channels in humid climates are characterized by
a discharge distribution with larger mean discharge but a lower coefficient of variation.
Conversely, channels in arid environments are characterized by lower mean annual
discharge but higher coefficients of variation. If the threshold for bed material
entrainment is relatively low then the sediment flux contribution of more frequent,
smaller discharges allows channels in humid climates to transport a greater quantity of
sediment in a given time period than similar channels located in arid climates.
Conversely, the return period for the effective discharge is negatively correlated with
mean annual runoff. The threshold for sediment entrainment and cross-sectional
geometry greatly affect the long-term sediment flux and effective discharge, and
therefore, gravel- and sand-bed channels are predicted to evolve over different time
scales in response to climate change.
216
Notation
b
power-law coefficient for the relationship between w and Q, s1/2/m1/2 if Q has
units of m3/s and the exponent is 1/2
Cf
dimensionless drag coefficient
Cv,Q
coefficient of variation of discharge
Cv,Qd
coefficient of variation of daily discharge
Cv,τ*
coefficient of variation of Shields stress
d
grain size of bedload sediment, m
ε
constant for scaling Shields stress with respect to Critical Shields stress
g
gravitational acceleration, m/s2
h
elevation, m
κ
diffusivity, m2/yr
μlnQ
mean of the natural log of discharge per unit channel width, m2/s
μlnτ*
mean of the natural log of Shields stress
n
Manning’s roughness coefficient
η
sediment transport equation exponent
qs*
non-dimensional sediment flux
q s*
long-term non-dimensional sediment flux
Q
volumetric discharge, m3/s
Q
mean annual discharge, m3/s
Qcf
channel-forming discharge, m3/s
217
Qe
effective discharge, m3/s
R
mean annual runoff, m
ρs
density of bedload sediment, kg/m3
ρf
density of fluid, kg/m3
S
local channel slope
S0
average channel slope for a given reach of simulated channel profile
σlnQ
standard deviation of the natural log of discharge per unit channel width, m3/s
σlnτ*
standard deviation of the natural log of Shields stress
t
time, s
TQe
effective discharge return period, yr
τ*
Shields stress
τ* cf
channel-forming discharge Shields stress
τ
mean Shields stress
*
τ* cr
critical Shields stress of bedload sediment
x
distance from the model origin, m
w
channel width, m
218
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225
Tables:
Table 1: Calculated and reported Diffusivities for channels draining in the Dead Sea
Number
Channel
Drainage
Area
(km2)
Mean Q
(m3/s)
Cv,Q
Slope
Grain
Size
(m)
Channel
Width
(m)
1
Qumeran
47
0.238
1.42
0.35
0.008
50
2
Qidron
123
1.046
1.24
0.04c
0.016
60c
3
Darga
237
2.347
1.19
0.05
0.016
80
4
Qedem
12
0.044
1.55
0.1
0.004
60c
5
Ishai
2
0.076
0.83
0.04c
0.005
60c
6
Hever
175
1.494
1.24
0.04c
0.008
60c
7
Zeelim
253
1.506
1.36
0.02
0.008
60c
a
Calculated
Diffusivity
(m2/day)
4.55a,
24.78b
35.11a,
3.18b
332.68a,
12.22b
4.42a,
112.99b
0.02a,
0.02b
617.2a,
140.48b
245.82a,
49.58b
Reported
Diffusivity
(m2/day)
8
11.25
22.5
1.25
0.041
6.75
1.5
Diffusivities calculated using rainfall index to predict mean annual runoff
Diffusivities calculated using a power-law relationship between mean annual runoff and
drainage area with an exponent of −0.5
c
Data was not explicitly provided by either Moshe et al. (2008) or Bowman et al. (2007)
b
Table 2: Parameters used in calculations for Figures 3, 4 and 6
Parameters
Drainage Area (km2)
Manning’s Roughness Coefficient, n
Grain Size, d (m)
Submerged Specific Gravity
Critical Shields Stress, τ*cr
Channel-Forming Discharge Shields Stress, τ*
Channel-Forming Discharge Return Period (yr)
Gravel-bed
1.0x104
0.035
0.02
1.7
0.047
0.07
1.5, 6.0
Sand-bed
1.0x104
0.035
0.001
1.7
NA
1.0
1.5, 6.0
Table 3: Calculated diffusivities and corresponding parameters
Drainage Area
(km2)
1.0x103
1.0x103
1.0x104
1.0x104
1.0x103
R
(m/yr)
1
0.1
1
0.1
1
Mean Q
(m3/s)
31.71
3.17
317.10
31.71
31.71
Cv,Q
0.868
1.616
0.868
1.616
0.868
d
(m)
0.01
0.01
0.01
0.01
0.001
S0
1.0x10-3
1.0x10-4
1.0x10-3
1.0x10-4
1.0x10-3
Diffusivities
(m2/yr)
1.09x104
5.34x101
2.67x106
5.59x104
1.27x105
226
1.0x103
1.0x104
1.0x104
0.1
1
0.1
3.17
317.10
31.71
1.616
0.868
1.616
0.001
0.001
0.001
1.0x10-4
1.0x10-3
1.0x10-4
2.11x104
7.13x105
1.19x105
227
Figures:
Figure 1: Relationship between mean annual runoff and the coefficient of variation of
daily discharge for 530 United States channels. Channels that drain areas between 10 and
5,000 km2 (gray) and 5,000 to 3.0x106 km2 (black) are plotted as asterisks. Linear
regression lines for this study are the following: (solid line) trend line for data binned by
mean annual runoff, (dashed line) trend line for all data points that represent drainage
basins larger than 5,000 km2 and (dotted line) trend line for all data points.
228
Figure 2: A comparison of diffusivity values for sand- (dashed lines) and gravel- (solid
thin lines represent cohesive channels and dotted-solid lines represent self-regulating
channels) bed channels as a function of mean Shields stress. Diffusivity values were
calculated for three coefficient of variation of Shields stress values that span the expected
range for U.S. channels (i.e., 0.2, 0.4 and 0.8).
229
Figure 3: Relationship between climate and long-term bedload sediment flux for gravel(A, C) and sand-bed channels (B, D). Each gray curve represents a different coefficient of
variation of daily discharge (i.e., 0.5, 2 and 8). For the gravel-dominated channels the
solid lines and black numbers represent cohesive channels. The dashed lines and gray
numbers represent self-forming channels. The dash-dotted lines in each figure represent a
channel in which the coefficient of variation of daily discharge is a function of mean
annual runoff. Figures (A, B) represent channels with slopes calibrated to the 1.5 year
230
discharge and Figures (C, D) represent channels with slopes calibrated to the 6.0 year
discharge.
Figure 4: The relationship between long-term sediment flux and climate (expressed in
terms of mean and coefficient of variation of discharge) for gravel- (A) and sand- (B) bed
channels of constant grain size and slope. The thick, white line represents the covariation
between mean annual runoff and the coefficient of variation of daily discharge observed
for modern U.S. channels.
231
Figure 5: A comparison of long-term bedload sediment transport rates for gravel-bed
channels that were calculated using a site-calibrated rating curve (solid line), the Barry et
al. (2004) transport formula (triangles), and the Wiberg and Smith (1989) transport
formula (asterisks).
232
Figure 6: The relationship between the effective discharge return period and parameters
of the hydroclimatology for gravel- (A, B, D) and sand- (B, D) bed channels. Figure 6A
represents a channel with cohesive banks and 6B represents a self-forming channel. The
thin, white line in (A, B, C) represents the covariation between mean annual runoff and
the coefficient of variation of daily discharge that is observed in modern channels. In
Figure 6D the dashed lines represent channels with slopes calibrated to the 6.0 year
discharge and the solid lines represent calibrating slope to the 1.5 year discharge.
233
Figure 7: A comparison of the effective discharge (A) and effective discharge return
period (B) predicted by our analytic solution to field observations for gravel-bed channels
from Barry et al. (2008) (triangles) and Torizzo and Pitlick (2004) (circles).